Advanced Quantum Theory - Sebastian...

Advanced Quantum Theory

Lecture Notes

Sebastian Muller

2020/21, Teaching block 2

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Contents

1 Basics 1

1.1 Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Feynman path integral 6

2.1 Propagator for short times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Propagator for arbitrary times . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Example: Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Elastic chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Path integrals in Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . 16

3 Perturbation theory 19

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Anharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Second quantisation 33

4.1 Two particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 N particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . 37

4.5 Hamiltonian in second quantisation . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.6 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Path integrals in second quantisation 45

5.1 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Fermions (not examinable) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Chapter 1

Basics

We start with some important background material in classical and quantum mechanics.

1.1 Classical mechanics

Lagrangian mechanics

Compared to Newtonian mechanics, Lagrangian mechanics has the advantage that we can usearbitrary sets of generalised coordinates to describe our system and our fundamental equationsalways have the same form. Let us denote the coordinates by q1, q2, . . . , qn and assemble theminto a vector q. Then the motion of the system is described by the function q(t). We nowneed the Lagrangian which the difference of the kinetic energy T and the potential energy Uexpressed as a function of q, q, and time,

L(q, q, t) = T − U.

Then the equations of motion are∂L

∂qα=

d

dt

∂L

∂qα.

To give an interpretation of the Lagrange equations we define the action of a trajectory (fol-lowed between times t1 and t2) as

S[q] =

∫ t2

t1

L(q(t), q(t), t)dt.

Here the square brackets highlight that the argument of S is a function. Now one can show thatthe Lagrange equations are equivalent to demanding that S[q] is stationary w.r.t. variationsof the function q(t) that preserve the boundary conditions, i.e., the values of q(t) at t = t1 andt = t2.

Hamiltonian mechanics

Hamiltonian mechanics is formulated in phase space, i.e., the fundamental variables are q aswell as the momenta defined by

p =∂L

∂q.

1

2 CHAPTER 1. BASICS

Instead of the Lagrangian we consider the Hamiltonian

H(q,p, t) = p · q − L .

The corresponding equations of motion are

qα =∂H

∂pα, pα = −∂H

∂qα.

1.2 Quantum mechanics

In quantum mechanics the variables qα, pα are replaced by operators qα, pα acting on wave-functions. In position representation the operator qα simply amounts to multiplication of thewavefunction with its parameter qα and the momentum operator is pα = ~

i∂∂qα

. The dynamicsof the wavefunctions is then given by the Schrodinger equation

Hψ(q, t) = i~∂

∂tψ(q, t)

where the quantum mechanical Hamiltonian is obtained from the classical Hamiltonian byreplacing all q’s and p’s by operators. As we will usually use Cartesian coordinates we willreplace q by r from now on.

We now go on to highlight some aspects of quantum mechanics that will be important for thiscourse.

Dirac notation

In bra-ket or Dirac notation wavefunctions ψ(r) are denoted by ”kets” |ψ〉. The scalar productof two wavefunctions φ(r) and ψ(r) is then denoted by∫

Rnφ∗(r)ψ(r)dnr = 〈φ|ψ〉.

I.e. the ”bra” 〈φ| applied to a wavefunction indicates multiplication with the complex conjugateof the φ(r) and subsequent integration.

When dealing with eigenstates in bra-ket notation, one sometimes writes the correspondingeigenvalue in the bracket.

The delta ”function”

Definition. The ”function” δ(r) is informally defined as follows:

• It vanishes for r 6= 0.

• It diverges at r = 0.

• If we integrate over it multiplied with a different function f(r) we get∫Rnf(r)δ(r)dnr = f(0).

1.2. QUANTUM MECHANICS 3

These properties also imply∫f(r)δ(r − r′)dnr = f(r′). A mathematically respectable way of

defining delta ”functions” is based on the theory of distributions1 however it is convenient towork with them as with usual functions.

Position eigenfunctions. δ(r) can be interpreted as a wavefunction. As it is nonvanishingonly for r = 0 a particle with this wavefunction is localised at the origin. In other words, δ(r)is an eigenfunction of the position operator with eigenvalue zero.

Similarly δ(r − r′) is nonvanishing only for r = r′ and forms an eigenfunction of the positionoperator with eigenvalue r′.2 It can also be written as the ket |r′〉.

We note that the position eigenfunctions are a bit different from other eigenfunctions as theyare distributions rather than proper functions, and there is one for every possible r so they donot form a discrete set parametrised by integers. One consequence of this is that they cannotbe normalised to have 〈r|r〉 = 1.

Application of 〈r′|. Applying the bra 〈r′| to a quantum state |ψ〉 simply means to writedown the wavefunction at position r′ as

〈r′|ψ〉 =

∫δ(r − r′)ψ(r)dnr = ψ(r′).

Integral representation of the delta function. The delta function can be obtained fromthe following integral (which will be considered in an exercise)

1

2π~

∫ ∞−∞

eipr/~dp = δ(r).

1Remark (not examinable). Distributions are mappings from a space of test functions F to C. One oftenuses

F = {f ∈ C∞(Rn) : (1 + |r|2)kf(r) is bounded for all k ∈ N}.Functions g : Rn → C give rise to distributions

Dg : F → R

Dg[f ] =

∫Rn

f(r)g(r)dnr ∈ R (1.1)

where f ∈ F . The δ-distribution maps each function f ∈ F to its value at zero

Dδ : F → CDδ[f ] = f(0) .

We now use the following notation motivated by (1.1)

Dδ[f ] =

∫Rn

f(r)δ(r)dnr .

We thus work with the δ-distribution as we would work with a function.2To see this formally, first for one dimensional systems, we note that an eigenfunction of the position operator

with eigenvalue r′ has to satisfyrψ(r) = r′ψ(r).

This is solved by ψ(r) = δ(r − r′) as

rδ(r − r′) = rδ(r − r′) = r′δ(r − r′) .

Here the second equality is trivially correct if r = r′. It is also correct if r 6= r′ as in that case δ(r − r′) = 0.The same argument can be used for vector valued r.

4 CHAPTER 1. BASICS

The analogous formula for r ∈ Rn is

1

(2π~)n

∫Rneip·r/~dnp = δ(r). (1.2)

Resolution of the identity

If the states |m〉 form a discrete orthonormal basis it was shown in 3rd year Quantum Mechanicsthat the identity operator can be written as

1 =∑m

|m〉〈m|.

We will later need an analogous result for position eigenstates. As these states form a contin-uous rather than a discrete basis the result for this case involves an integral:

1 =

∫|r〉〈r|dnr (1.3)

To prove this result we consider matrix elements where the r.h.s. is sandwiched between twoarbitrary states 〈φ| and |ψ〉. We then have

〈φ|(∫|r〉〈r|dnr

)|ψ〉 =

∫〈φ|r〉〈r|ψ〉dnr =

∫φ(r)∗ψ(r)dnr = 〈φ|ψ〉 = 〈φ|1|ψ〉

proving our claim.

Momentum eigenfunctions

We also need eigenfunctions of the momentum operator. These are given by

ψp(r) =1

(2π~)n/2ei~p·r

.

To see that these are eigenfunctions, we apply to momentum operator and get

pψp(r) =~i

∂

∂rψp(r) =

~i

i

~pψp(r) = pψp(r).

The prefactor in ψp(r) was chosen to make sure that there is a resolution of the identity resultanalogous to position eigenfunctions,

1 =

∫|p〉〈p|dnp. (1.4)

This result can be proved by evaluating the r.h.s. sandwiched between arbitary position eigen-functions 〈r| and arbitary states |ψ〉,

1.2. QUANTUM MECHANICS 5

〈r|∫dnp|p〉〈p|ψ〉 =

∫dnp

1

(2π~)n/2ei~p·r

∫dnr′

1

(2π~)n/2e−

i~p·r

′ψ(r′)

=

∫dnp

∫dnr′

1

(2π~)nei~p·(r−r

′)ψ(r′)

=

∫dnr′ δ(r − r′)ψ(r′)

= ψ(r) = 〈r|ψ〉 = 〈r|1|ψ〉

where we use (1.2) in the third line.

Time evolution in quantum mechanics

A formal solution of the Schrodinger equation

Hψ(r, t) = i~∂

∂tψ(r, t)

is

ψ(r, t) = e−i~ Htψ(r, 0) (1.5)

This result is obtained heuristically if we momentarily forget that H is an operator and weintegrate as if it were, say, a real number. If we want the result to be true for operatorswe first of all have to define what an exponential of an operator means. We can define suchexponentials through the Taylor series of the the exponential, as in

e−i~ Ht =

∞∑n=0

1

n!

(− i~Ht

)n.

Here Hn is the operator obtained by applying H n times. e−i~ Ht is also referred to as the time

evolution operator.

Proof: Now we prove (1.5). Using that the derivative of the operator exponential defined aboveobeys the same rules as the derivative of a usual exponential we get

i~∂

∂t

(e−

i~ Htψ(r, 0)

)= i~

(− i~H

)(e−

i~ Htψ(r, 0)

)= H

(e−

i~ Htψ(r, 0)

)which means that the Schrodinger equation is satisfied. Moreover the claimed formula forψ(r, t) reduces to ψ(r, 0) if we insert t = 0.

Propagator. To describe the time evolution of a quantum system it is helpful to considerthe propagator, defined by the matrix elements of the time evolution operator,

K(rf , r0, t) = 〈rf |e−i~ Ht|r0〉.

The propagator represents a state starting with |r0〉, i.e., a delta function located at r0, attime 0. Then this state evolves according to the Schrodinger equation over a time interval t, as

expressed by the time evolution operator e−i~ Ht. Applying 〈rf | means that we are considering

the resulting wavefunction at the position rf .

Chapter 2

Feynman path integral

The Feynman path integral provides a way to express quantum mechanics in terms of trajec-tories reminding of classical mechanics. Specifically it expresses the propagator as

K(rf , r0, t) =

∫D[r]e

i~S[r] (2.1)

Here the integral is taken over all trajectories that go from from r0 to rf during the time t.These trajectories are described by functions r(t′) with r(0) = r0 and r(t) = rf . Hence wehave functions as integration variables! This is unusual and we we will give a proper definitionof such integrals including the integration measure D[r] only at a later stage.

The integrand, i.e., the weight attributed to each trajectory, depends on the action of eachtrajectory.

We stress that the integral is taken over all trajectories from r0 to rf , not just those obeyingthe classical laws of motion. In hindsight this is not surprising as quantum mechanics goesbeyond classical mechanics so its formulation should involve elements not present in classicalmechanics. However we still expect that in some sense the contributions from trajectoriesobeying the classical laws of motion should dominate. We will come back to this point at alater stage.

2.1 Propagator for short times

We start by deriving the path integral. We first consider the case of short times t and thenproceed to arbitrary times. We will concentrate on systems with a Hamiltonian of the form

H =p2

2m︸︷︷︸=T

+U(r) ,

i.e. a standard kinetic energy plus potential. The propagator is then given by

K(rf , r0, t) = 〈rf |e−i~ Ht|r0〉

where e−i~ Ht = e−

i~ (T+U)t. It would be very nice if we could write this as e−

i~ T te−

i~ Ut as appli-

cation of the exponentiated potential to the position eigenstate |r0〉 would simply amount to

6

2.1. PROPAGATOR FOR SHORT TIMES 7

multiplication with e−i~U(r0)t. However in general it is not correct to replace an exponentiated

sum of operators by the product of the exponentiated summands. (Instead one has to use theso-called Baker-Campbell-Hausdorff formula which also involves the commutator if the twooperators.) However in the present case we are only interested in small t, and we get throughwith the above replacement up to corrections of quadratic and higher orders in t. We have

〈rf |e−i~ Ht|r0〉 ≈ 〈rf |

(1− i

~Ht

)|r0〉

≈ 〈rf |(

1− i

~T t

)(1− i

~U t

)|r0〉

≈ 〈rf |e−i~ T t e−

i~ Ut|r0〉︸︷︷︸

=e−i~U(r0)t|r0〉

= 〈rf |e−i~ T t|r0〉︸︷︷︸

=:A

e−i~U(r0)t (2.2)

Here we used first used a Taylor expansion, then we included a negligible error proportionalto T U t2, then we made a Taylor expansion ’in reverse’, and finally we just applied to operatorU .

Evaluation of A. When evaluating A, it is helpful to apply T to momentum eigenfunctionsinstead of position eigenfunctions. Hence we insert a resolution of the identity in terms ofmomentum eigenfunctions. This leads to

A = 〈rf |e−i~ T t

∫dnp|p〉〈p|r0〉

=

∫dnp〈rf |e−

i~ T t|p〉〈p|r0〉

=

∫dnp〈rf |e−

i~

p2

2m |p〉〈p|r0〉

=

∫dnp e−

i~

p2

2m 〈rf |p〉︸︷︷︸1

(2π~)n/2ei~p·rf

〈p|r0〉︸︷︷︸1

(2π~)n/2e−

i~p·r0

=1

(2π~)n

∫Rndnp e−

i~

p2

2mteip·(rf−r0)/~

=

n∏k=1

1

2π~

∫ ∞−∞

dpke− i

~p2k2m

t+ipk(rfk−r0k)/~.

Here we have used that p|p〉 = p|p〉, as well as the explicit formula for the momentum eigen-functions. If we were working in Hamiltonian mechanics it would be best to keep the momentumintegral and also include it in the path integral, but in Lagrangian mechanics we evaluate theintegral instead. It is quite similar to a Gaussian integral and can be done with the followingresults about integrals of this type:

• The Gauss integral gives ∫ ∞−∞

e−ax2dx =

√π

a

for a > 0.

8 CHAPTER 2. FEYNMAN PATH INTEGRAL

• The Fresnel integral involves an additional i and is evaluated as∫ ∞−∞

e∓iax2dx =

√π

±ia=

√π

|a|e∓i

π4

sign a. (2.3)

Heuristically these results follow from the Gauss integral by replacing a → ±ia, usingi = ei

π2 , and hoping for the best. The rigorous derivation involves contour integration in

the complex plane.

• Our pk-integral also involves linear terms in the exponent. Hence we have to evaluate ashifted Fresnel integral

∫∞−∞ e

−iax2+ibxdx. By completing the square as in

−iax2 + ibx = −ia(x− b

2a

)2

+ iab2

4a2

and substituting y = x− b2a this integral can be evaluated as∫ ∞

−∞e−iax

2+ibxdx =

∫ ∞−∞

e−iay2dy ei

b2

4a =

√π

iaeib2

4a . (2.4)

If we apply (2.4) to (??) with a = t2m~ , b =

rfk−rik~ we finally get

A =

n∏k=1

( m

2πi~t

)1/2exp

(i

~1

2m

(rfk − r0k

t

)2

t

).

After simplification and combination with (2.2) this gives our final result for the short-timepropagator:

K(rf , r0, t) ≈( m

2πi~t

)n/2exp

(i

~

[1

2m

(rf − r0

t

)2

− U(r0)

]t

)(2.5)

This result is reassuring given our final goal (2.1): If we moved with constant velocity from r0

to rf in time t that velocity would be given byrf−r0

t . For a different (differentiable) motionbetween these points it is still a good approximation of the velocity for small times t. Hencethe expression in the square brackets is a good approximation for the Lagrangian L = T − U .Multiplied with t we get a good approximation for action over a small time interval duringwhich the Lagrangian does not change much.

2.2 Propagator for arbitrary times

The propagator for arbitrary times t can be evaluated using our preceding result if we splitthe time interval t into N intervals of size τ = t

N . If we then apply the limit N → ∞ we canmake use of the result for small times. We thus write

K(rf , r0, t) = 〈rf |e−i~ Hτ . . . e−

i~ Hτ |r0〉

(with N factors). We now insert resolutions of the identity 1 =∫|r〉〈r|dnr between the factors.

As there are N factors we need N − 1 integrals. We thus obtain

K(rf , ri, t) =

∫. . .

∫〈rf |e−

i~ Hτ |rN−1〉〈rN |e−

i~ Hτ |rN−2〉〈rN−2| . . . |r2〉

〈r2|e−i~ Hτ |r1〉〈r1|e−

i~ Hτ |r0〉 dnr1 . . . d

nrN−1.

2.2. PROPAGATOR FOR ARBITRARY TIMES 9

To simplify notation in the following it is helpful to set rN = rf ; however like r0, rN willremain fixed and is not an integration variable. Now we can use (2.5) for all N factors. Foreach of them we obtain one power of the prefactor in (2.5) with time t

N ; the exponents are alladded and in these exponents the positions turn into rj and rj+1. We thus obtain

K(rf , r0, t) =

∫dnr1 . . . d

nrN−1

(mN

2πi~t

)nN/2exp

i

~

N−1∑j=0

[1

2m

(rj+1 − rj

τ

)2

− U(rj)

]τ

We can now imagine the points rj to arise from a discretisation of a trajectory, with rj beingthe position at time jτ , see Fig. 2.1. (This is consistent with our derivation as the resolution

of the identity with integration variable rj was inserted after j factors e−i~ Hτ each describing

evolution over a time τ .) Then, given our earlier discussion, 12m(rj+1−rj

τ

)2−U(rj) is a good

approximation for the Lagrangian during the motion from rj to rj+1, and its product with τis a good approximation for the action associated to that piece of trajectory. The sum over jleads to a summation over all pieces of trajectory and hence to a good approximation for theoverall action. The approximation gets better and better the finer we make our discretisation,i.e., the larger we make N . In the limit N →∞ we indeed obtain the action,

N−1∑j=0

[1

2m

(rj+1 − rj

τ

)2

− U(rj)

]τ →

∫ t

0

(1

2mr(t′)2 − U(r(t′))

)dt′ = S[r]

(N →∞)

Figure 2.1: A discretised trajectory.

Now it is important that we integrate over r1, . . . , rN−1. I.e. we take into account all discretisedversions of trajectories, and in the limit N →∞, we take into account all possible trajectoriesor in other words all functions r(t′). The only restriction is that we have to start at r0 andend at rN = rf . For our function this implies the anticipated boundary conditions r(0) = r0


and r(t) = rf . There are no further restrictions on these function, in particular they don’thave to be solutions for the classical equations of motion.

If we simply define our integration measure on the space of functions by the limit

∫D[r] . . . = lim

N→∞

(mN

2πi~t

)nN/2 ∫dnr1 . . . d

nrN−1 . . .

we get the expected path integral

K(rf , r0, t) =

∫r(0)=r0,r(t)=rf

D[r]ei~S[r]. (2.6)

Remarks.

• The Feynman path integral provides a link between quantum mechanics and tra-jectories which are the essential objects in classical mechanics. As it also involvestrajectories that don’t obey the classical laws of motion it goes beyond classical mechan-ics. This is to be expected as quantum mechanics is the deeper theory and should involvecorrections to classical behaviour.

• However trajectories satisfying the classical laws of motion dominate in thefollowing sense: These trajectories are stationary points of the action. Hence varyinga trajectory compared to a classical one does not change the action to linear orderin the deviation. Changes to the original action appear only in quadratic or higherorders. So trajectories close to a classical one have very similar actions and give verysimilar contributions to the path integral. By contrast the actions of trajectories furtheraway from a classical one already show deviations from each other already in a linearapproximation. As an action difference of π~ already leads to a change in sign thismeans that the contributions of such trajectories have a tendency to cancel. This leavescontributions close to classical trajectories as the dominant ones.

One can use this idea for approximations of the path integral in terms of classical tra-jectories. These approximations are used e.g. in Quantum Chaos.

• For a rigorous theory one would have to pay more attention to the definition of the inte-gration measure and to trajectories that are not differentiable. The rigorous formulationof path integrals is still not complete.

• We assumed that all positions r ∈ Rn are possible. If some r are ruled out and this leadsto a ’not simply connected’ position space (for example a two dimensional space with ahole) a similar result applies. But there is an additional phase factor depending on howoften each trajectory winds around the hole.

• In our derivation we introduced an integral over p in order to represent a delta function,but we got rid of this integral again because it was a shifted Fresnel integral that couldbe evaluated explicitly. However, one could have just left this integral. As it appearedin the formula for the short time propagator one would then obtain momentum integralsfor each of the N time steps in the propagator for arbitrary times. These integrals can

2.3. EXAMPLE: HARMONIC OSCILLATOR 11

be treated very similarly to the integrals in position space, so that in the end we wouldobtain a path integral in phase space with integrations over functions r and p. Thisintegral, to be derived in an exercise, has the form

K(rf , r0, t) =

∫D[r]D[p] exp

(i

~

∫ t

0(p(t′) · r(t′)−H(r(t′),p(t′))dt′

)where the integration measure

∫D[r]D[p] . . . is different from the one in the position

space path integral. The exponent involves the natural generalisation of the action tophase space, using that H = p · r − L ⇒ L = p · r −H.

2.3 Example: Harmonic oscillator

As an example we want to evaluate the path integral for the (one-dimensional) harmonicoscillator with the Lagrangian

L =1

2mx2 − 1

2mω2x2 .

The action is

S[x] =

∫ t

0

(1

2mx(t′)2 − 1

2mω2x(t′)2

)dt′.

It is helpful to split x(t′) into the classical solution (the stationary point of the action subjectto our boundary conditions x(0) = x0, x(t) = xf ) xcl(t

′) and the deviation from the classicalsolution δx(t′),

x(t′) = xcl(t′) + δx(t′).

As xcl satisfies the same boundary conditions as x the deviation δx must vanish at the bound-aries, δx(0) = δx(t) = 0. If we insert this decomposition of x into S[x] we get

S[x] =

∫ t

0

(1

2mxcl(t

′)2 − 1

2mω2xcl(t

′)2

)dt′

+

∫ t

0

(mxcl(t

′) ˙δx(t′)−mω2xcl(t′)δx(t′)

)dt′

+

∫ t

0

(1

2m ˙δx(t′)2 − 1

2mω2δx(t′)2

)dt′.

Here we have arranged the terms as in a Taylor expansion in δx: the first line contains termsindependent of δx (i.e. the action evaluated at xcl), the second line contains linear terms, andthe third line contains quadratic terms. Now it is important that we are expanding arounda stationary point of the action. As always, in a Taylor expansion around a stationary pointthe linear term vanishes because the derivative1 vanishes. In the special case of the harmonicoscillator it also turns out that the quadratic term can just be written as S[δx]. We thus have

S[x] = S[xcl] + S[δx].

Classical solution. To proceed we compute S[xcl]. xcl(t′) is the solution of Lagrange’s

equation,∂L

∂x=

d

dt′∂L

∂x

1In our case this is a functional derivative as discussed in Mechanics 2/23.


which for the harmonic oscillator boils down to

−mω2x2cl = mxcl.

Hence we have

xcl(t′) = A sinωt′ +B cosωt′.

If we insert this result into the action (2.3) we obtain

S[xcl] =mω

2

((A2 −B2)

sin 2ωt

2+ 2AB

cos 2ωt− 1

2

).

To satisfy the boundary condition at 0 we need xcl(0) = B = x0. The boundary condition att is satisfied if

xcl(t) = A sinωt+B cosωt = xf ⇒ A =xf −B cosωt

sinωt=xf − x0 cosωt

sinωt

Inserting these values into S[xcl] gives

S[xcl] =mω

2 sinωt

((x2

0 + x2f ) cosωt− 2x0xf

).

Integration measure. We now have the intermediate result

K(xf , x0, t) =

∫D[x]e

i~ (S[xcl]+S[δx]) (2.7)

with S[xcl] as above. It would be very nice if we could replace D[x] by D[δx]. This is permittedbecause it just amounts to a shift of the integration variable by a constant amount, but becausewe are dealing with a functional integral we want to work it out explicitly. We have∫

D[x] . . . = limN→∞

(mN

2πi~t

)nN/2 ∫dx1

∫dx2 . . .

∫dxN−1 . . .

= limN→∞

(mN

2πi~t

)nN/2 ∫dδx1

∫dδx2 . . .

∫dδxN−1 . . .

=

∫D[δx] . . .

Here we made, for each step j, the variable substitution xj → δxj where xj = xcl,j + δxj . Asxcl,j is not an integration variable the Jacobian for this transformation is 1.

As noted earlier the boundary conditions for δx are δx(0) = δx(t) = 0. After the variabletransformation the propagator can thus be written as

K(xf , x0, t) =

(∫δx(0)=δx(t)=0

D[δx]ei~S[δx]

)︸︷︷︸

=A(t)

ei~S[xcl]

= A(t) exp

(i

~mω

2 sinωt

((x2


)). (2.8)

Here the δx integral does not depend on x0 and xf at all (not even through the boundaryconditions) but only on t.

2.3. EXAMPLE: HARMONIC OSCILLATOR 13

How to determine A(t)? To get A(t) we could use the discretised version of the path inte-gral and integrate over δx1, δx2, . . . but this calculation becomes a little messy – this methodworks better for the propagator with vanishing potential which we are leaving for the home-work. Instead we are employing a shortcut to determine A(t), and we will only consider thecase 0 < ωt < π.2

Using the resolution of the identity we can derive the following result for integrals of productsof propagators ∫

K(xf , x, t2)K(x, x0, t1)dx

=

∫〈xf |e−

i~ Ht2 |x〉〈x|e−

i~ Ht1 |x0〉dx

= 〈xf |e−i~ H(t1+t2)|x0〉

= K(xf , x0, t1 + t2).

Intuitively this result means that time evolution according to the Schrodinger equation overtime t1 followed by time evolution over time t2 gives time evolution over the combined timet1 + t2. For our present argument we just need the special case x0 = xf = 0,∫

K(0, x, t2)K(x, 0, t1)dx = K(0, 0, t1 + t2). (2.9)

Now the A(t) in (2.8) has to be picked such that (2.9) is satisfied. The r.h.s. gives A(t1 + t2).For the l.h.s. we obtain∫

K(0, x, t2)K(x, 0, t1)dx

=

∫A(t2) exp

(i

~mω

2 sinωt2x2 cosωt2

)A(t1) exp

(i

~mω

2 sinωt1x2 cosωt1

)= A(t1)A(t2)

∫exp

(i

~mω

2

sinωt1 cosωt2 + sinωt2 cosωt1sinωt1 sinωt2

x2

)dx

= A(t1)A(t2)

√2π~mω

sinωt1 sinωt2sinω(t1 + t2)

eiπ4

where in the final step we used a trigonometric identity and the Fresnel integral. Now theequation can be rearranged into terms associated to the times t1, t2, and t1 + t2. If we do this,

and also multiply through with√

2π~mω sinω(t1 + t2)ei

π4 , we obtain√

2π~mω

sinω(t1 + t2)eiπ4A(t1 + t2)

=

(√2π~mω

sinωt1eiπ

4A(t1)

)(√2π~mω

sinωt2eiπ

4A(t2)

)

The easiest way to satisfy this equation is to set the l.h.s., the two factors on the r.h.s., andany term of the same form, to 1. This means that for all times t under consideration,√

2π~mω

sinωteiπ4A(t) ⇒ A(t) =

√mω

2π~ sinωte−i

π4 (2.10)

2Actually there is even a third way to do this calculation. This is based on a Fourier transform and brieflyoutlined in the Altland & Simons book.


and hence

K(xf , x0, t) =

√mω

2π~ sinωtexp

(i

~mω

2 sinωt

((x2


)− iπ

4

)

Note that we have just seen that our choice of A(t) is the simplest way to satisfy (2.10) butwe did not look for other solutions. Hence to be really sure one still has to plug our resultinto the Schrodinger equation and check that it works – indeed it does. (This is shown in anexercise.)

We have just considered 0 < ωt < π as in this case sinωt is positive. For arbitrary ωt one hasto be more careful with the phase factor appearing in the Fresnel integral (a complication thatis often ignored in the literature).

2.4 Elastic chain

We want to present a further example for a path integral. An interesting feature of this pathintegral is that in a certain limit it allows for a representation in terms of a field. This willserve as a simple illustration for the use of path integrals in the context of fields; more involveduses of path integrals with fields originate from quantum field theory and condensed mattertheory.

We consider a chain of N+1 masses m connected by springs with spring constant k and naturallength 0. We take the length of the chain as C. The positions along the chain will be denotedby Φi. We require that the initial mass is fixed at Φ0 = 0 and the final one is fixed at ΦN = C.The positions Φi of the remaining masses (i = 1 . . . N − 1) will be assembled into a vector Φ.Then the Lagrangian is

L(Φ, Φ) =N−1∑i=1

1

2mΦ2

i −N−1∑i=0

1

2k(Φi+1 − Φi)

2. (2.11)

The vector Φ can now be treated just like the position r considered earlier and hence we obtainthe propagator of our system as the path integral

K(Φf ,Φ0, t) = 〈Φf |e−i~ Ht|Φ0〉 =

∫Φ(0)=Φ0,Φ(t)=Φf

D[Φ]ei~S[Φ].

This propagator describes the time evolution of a quantum mechanical chain with springs. Thestates |Φ0〉 and |Φf 〉 are eigenstates where the positions of all the masses in the system arefixed and K(Φf ,Φ0, t) gives the wavefunction that started with initial condition Φ0〉 and isthen evaluated at Φf after time evolution over time t. The integral goes over all Φ(t′) withthe appropriate conditions at 0 and t, and we have

S[Φ] =

∫ t

0dt′L(Φ(t′), Φ(t′)). (2.12)

Continuum limit

We are now interested in the limit where the length C stays fixed but the number of massesand springs N is taken to infinity. This limit describes a string or rubber band whose different

2.4. ELASTIC CHAIN 15

parts can be compressed or expanded independently and are thus modelled by separate springsinstead of modelling the whole system by just one spring; the only constraint when expandingand contracting the different parts is that the total length has to stay fixed.

In our limit it is convenient to replace i by a continuous parameter. To choose a convenientparameter we use that at equilibrium the masses will be equally spaced with distance a = C

N ,such that each mass i has the position ia. We thus replace the index i as a parameter bythe equililibrium position x = ia. For finite N this means that Φ is parameterised by a realparameter that can still only assume discrete values a, 2a, . . . . However in the limit N →∞ itcan really assume continuous values. Φ thus becomes a function Φ(x, t′) of positions x as wellas times t′, or in other words a field.

It is natural to expect that the discrete sum over i in the Lagrangian will then also be replacedby an integral. Moreover we one would expect that the difference Φi+1 − Φi is replaced by aderivative w.r.t. x (denoted by a prime). This leads to

L(Φ, Φ) ≈∫ C

0dx

(1

2

m

aΦ(x, t′)2 − 1

2kaΦ′(x, t′)2

)(2.13)

where the approximation turns into an identity as N → ∞. Note that we have not yet givena derivation of the coefficients m

a and ka. This will be done in the end of this section. Withthe above result for the Lagrangian the path integral turns into

K(Φf ,Φ0, t) =

∫D[Φ]eiS[Φ]/~

where the action (2.12) can be written as a double integral

S[Φ] =

∫ C

0dx

∫ t

0dt′(

1

2

m

aΦ(x, t′)2 − 1

2kaΦ′(x, t′)2

)︸︷︷︸

=L(Φ,Φ,Φ′)

Here the initial and final conditions are given by functions Φ0(x) and Φf (x) and the integralis taken over fields Φ(x, t′) with

Φ(x, 0) = Φ0(x), Φ(x, t) = Φf (x)

for all x. In the action position and time are treated on the same footing, and it is natural tointerpret the integrand L as a Lagrangian density whose role is analogous to the one playedby the Lagrangian in usual Lagrangian mechanics. However note that the Lagrangian for theproblem is still (2.13). In addition to the dependence on the field (absent here) and its timederivative the Lagrangian density also depends on the spacial derivative. But time is stillsingled out as we consider time evolution and we take initial and final conditions at specifiedtimes 0 and t.

Further details (not examinable)

To derive (2.13) we show that each summand in L (see Eq. (2.11)) can be replaced by anintegral from ia to (i+ 1)a implying that the whole sum can be replaced by an integral from


0 to C. We start by making the replacement

Φi(t′) = Φ(ia, t′)

in the summand, leading to

1

2mΦ(ia, t′)2 − 1

2k (Φ((i+ 1)a, t′)− Φ(ia, t′))2︸︷︷︸

≈(Φ′(ia,t′)a)2

where we Taylor expanded denoting the derivative w.r.t. x by a prime. Now the summandcan be approximated further by

1

2

m

a

∫ (i+1)a

iadxΦ(x, t′)2 − 1

2ka

∫ (i+1)a

iadxΦ′(x, t)2

where we the integrands are almost constant in integration interval (exactly constant in thelimit a → 0). Hence integration only leads to multiplication with a which is compensated bydividing out a. Combining all summands we obtain (2.13) as desired. Here all occurrences ofindices i are replaced by the continuous variable x. Moreover our earlier requirements Φ0 = 0and ΦN = C now turn into the boundary conditions

Φ(0, t′) = 0

andΦ′(C, t′) = C

for all t′.

To keep the coefficients from going to zero and infinity as N → ∞ and a → 0 we have to letm go to zero at the same rate as a = C

N , and k has to diverge. Indeed this is the appropriatelimit for our physical system. In particular it means that the overall mass of the system staysfixed3 as it should if we take a given string or rubber band and just discretise it in a finer andfiner way.

2.5 Path integrals in Statistical Mechanics

Path integrals also have important applications in Statistical Mechanics. In this case it is notthe propagator that is written as a path integral, but a slightly different quantity that weare about to introduce. In Statistical Mechanics one is often interested in systems that canexchange energy with the environment. Hence the energy of the system is not fixed. One canshow that the statistical probability to find it in the j-th energy eigenstate is proportional to

e− 1kBT

Ej

where T is the temperature and kB is called the Botzmann constant; one often uses theabbreviation β = 1

kBT. In order to normalise these probabilities we need to compute the

partition function given by

Z =∑j

e−βEj .

3To be precise, if we choose M as the fixed overall mass we have m = MN+1

i.e. m shrinks like 1N+1

rather

than 1N

but the difference between these is negligible as N →∞.

2.5. PATH INTEGRALS IN STATISTICAL MECHANICS 17

The partition function can also be written as

Z = tr e−βH ,

using that the eigenvalues of e−βH are e−βEj and that the trace is the sum of the eigenvalues.

Now e−βH looks formally very similar to the time-evolution operator, and it is natural tolook for a corresponding path integral. One possibility is to determine a path integral for thepartition function itself, which involves a trace. However in order to increase the similaritywith what we have done before, we instead consider the matrix elements

〈rf |e−βH |r0〉.

We note that here rf and r0 could also be collections of the positions of many particles, andone can easily construct the trace from the matrix elements.

Path integral. The path integral for 〈rf |e−βH |r0〉 can be obtained from the path integral

for 〈rf |e−i~ Ht|r0〉 by replacing i

~ t by β. This is somewhat heuristic but the result is the sameas obtained with a more careful approach. Recalling that

〈rf |e−i~ Ht|r0〉 =

∫D[r] exp

[i

~

∫ t

0

(1

2mr(t′)2 − U(r(t′))

)dt′]

we also have to make an analogous change for the integration variable t′, replacing i~ t′ by β′

which runs from 0 to β. Then i~dt′ is replaced by dβ′, simplifying the exponential. This leads

to

〈rf |e−βH |r0〉 =

∫D[r] exp

[∫ β

0

(1

2m

(i

~dr

dβ′

)2

− U(r(β′))

)dβ′

]where in the kinetic energy we had to replace the time derivative d

dt′ (indicated by the dot) byi~ddβ′ . This result can be rewritten as

〈rf |e−βH |r0〉 =

∫D[r] exp

[−∫ β

0

(1

2m

(1

~dr

dβ′

)2

+ U(r(β′))

)︸︷︷︸

=SE [r]

dβ′

].

Here we have used that due to the squared factor i the kinetic energy contributes with anegative sign. It can be combined with the potential which contributed with a negative signanyway. The integrand is now (up to the scaling with ~) the energy, and its integral over theparameter β′ taking the role of time is called the Euclidian action.

Extracting the ground state energy. A nice feature of 〈rf |e−βH |r0〉 is that it allows to

determine the energy of the ground state. Using the spectral decomposition of e−βH in termsof energy levels Ej and energy eigenstates |ψj〉,

e−βH =∑j

e−βEj |ψj〉〈ψj |,

we can write〈rf |e−βH |r0〉 =

∑j

e−βEj 〈rf |ψj〉〈ψj |r0〉.


In the limit β →∞ all summands decrease exponentially but the summand corresponding tothe ground state decreases most slowly as this state has the lowest energy. Denoting the ground

state energy by E0 we see that for β →∞, 〈rf |e−βH |r0〉 is dominated by a term proportionalto e−βE0 .

We show for the harmonic oscillator how this can be used to determine E0. The propagatorfor the harmonic oscillator is

〈xf |e−i~ Ht|x0〉 =

√mω

2π~ sinωtexp

(i

~mω

2 sinωt((x2

0 + x2f ) cosωt− 2x0xf )− iπ

4

).

However we can work with the propagator for x0 = xf = 0,

〈0|e−i~ Ht|0〉 =

√mω

2π~ sinωtexp

(−iπ

4

).

Replacing i~ t by β gives

〈0|e−βH |0〉 =

√mω

2π~ sin(ω ~i β) exp

(−iπ

4

)∝

(sin

(ω~iβ

))−1/2

=

(eiω

~iβ − e−iω

~iβ

2i

)−1/2

In the limit β → ∞ the first term in the numerator dominates. In this limit the leadingcontribution is thus proportional to

(eω~β)−1/2 = e−β12~ω

which leads to a ground state energy of

E0 =1

2~ω.

This is of course in line with the result from third-year Quantum Mechanics. In the nextchapter we will come back to this way of computing ground state energies for the harmonicoscillator with a perturbation.

Chapter 3

Perturbation theory

3.1 Motivation

In the previous chapter we were able to evaluate path integrals explicitly for examples such asthe harmonic oscillator or (in the homework) a particle without potential. This was possiblebecause the Lagrangian or Hamiltonian in these cases contained terms up to quadratic orderin the relevant variables and then the integrals involved were Gauss or Fresnel integrals (orsimpler). However when they are terms of higher order than quadratic this is no longer possible.An approach that can treat this very frequent situation is provided by perturbation theory.Considering the Lagrangian version of path integrals for now, perturbation theory works well ifthe Lagrangian can be written as a term that contains contributions of at most quadratic orderin r and r, plus a small additional term (normally in the potential) that contains perturbationsof cubic or higher order. We then have

L(r, r) = L0(r, r)︸︷︷︸quadratic

+εL1(r, r).

Here ε is a small parameter, and the perturbation to the Lagrangian is given by εL1. Forexample we can consider L1 = r4. Now we can plug this Lagrangian into the path integral,leading to ∫

D[r]ei~∫Ldt′ =

∫D[r]e

i~∫L0dt′e

iε~∫L1dt′ ,

and we can Taylor expand the second exponential for small ε. If we go up to linear order in εwe obtain

eiε~∫L1dt′ ≈ 1 +

iε

~

∫L1dt

′.

The path integral thus obtains a leading contribution that is just the same as without pertur-bation, and the additional summand

iε

~

∫D[r]

(∫L1dt

′)ei~∫L0dt′ .

Evaluating this summand will lead to a Fresnel integral, as L0 is quadratic, but with ana prefactor in front of the exponential, e.g. involving L1 = r4. In order to be ableto evaluate path integrals with terms beyond quadratic order we thus have to be learn howto evaluate Fresnel integrals with prefactors. Further expressions of this type arise if we gobeyond linear order in expanding the exponential.

19

20 CHAPTER 3. PERTURBATION THEORY

Similar problems. There are many situations leading to the same problem, sometimes withGaussians instead of Fresnel integrals:

• The simplest integrals of a similar type do not involve any paths, just a Gaussian orFresnel integral with further factors outside the exponent, e.g.∫

xk1xk2 . . . e− 1

2x·Axdnx

where A is a matrix and x is a vector. We will look at these examples first. Theyillustrate the use of perturbation theory in the setting of matrices and vectors with dis-crete parameters, which is relevant for example in Random Matrix Theory and QuantumInformation.

• If we look for partition functions of systems with quadratic and higher-order contributionsto the potential we can split off the higher order contribution similarly to as for thepropagator. The formulas will be analogous apart from the replacement i

~ t→ β discussedbefore, and instead of the Fresnel integral we have a Gaussian integral.

• In Statistical Mechanics we could also be interested in evaluating statistical averages ofsome quantities, so even if the kinetic and potential energy are quadratic we could havea quantity in front of the Gaussian that we would like to average.

• In quantum field theory and particle physics so called correlation functions play an im-portant role. These lead to path integrals of the type∫

D[x]x(t1)x(t2)ei~∫Ldt′

which also require to consider Fresnel integrals with a prefactor even in case the La-grangian involves only quadratic terms.

3.2 Wick’s theorem

We want to start by considering integrals of the form∫xk1xk2 . . . exp

(−1

2xTAx

)dnx.

where x = (x1, x2, . . . , xn) is an n-dimensional real vector and A is an n × n real symmetrixmatrix. For the integral to converge we require A to be positive definite. This means thatxTAx must be positive for all x 6= 0, or equivalently all eigenvalues of A must be positive.

We start by evaluating the Gaussian integral without prefactors.

Lemma. ∫Rn

exp

(−1

2xTAx

)=

(2π)n/2√detA

≡ c (3.1)

This generalises the one-dimensional Gaussian integral∫

exp(−1

2ax2)dx =

√2πa .

3.2. WICK’S THEOREM 21

Proof. We start be showing the proposition for the case that A is a diagonal matrix withdiagonal elements a1, a2, . . . , an. We then have∫

exp

(−1

2xTAx

)dnx =

∫exp

(−1

2

n∑k=1

akx2k

)dnx =

n∏k=1

√2π

ak=

(2π)n/2√detA

.

Here we have evaluated the one-dimensional Gauss integrals for each k and then used that thedeterminant of a diagonal matrix is the product of its diagonal elements.

For the general case we use that every real symmetric matrix can be written as

A = OT

a1

. . .

an

︸︷︷︸

D

O

where O is an orthogonal matrix with OT = O−1 and a1, . . . , an are the eigenvalues forming adiagonal matrix D. With this representation of A the integral can be written as∫

exp

(−1

2xTAx

)dnx =

∫exp

(−1

2xTOTDOx

)dnx

=

∫exp

(−yTDy

)dny =

(2π)n/2√detD

=(2π)n/2√

detA

Here we defined y = Ox and transformed the integral into an integral over y. The Jacobianmatrix of this transformation is O which as an orthogonal matrix has a determinant withabsolute value 1. Finally we used that the detD is the product of the eigenvalues a1, . . . , anand hence equal to detA.

Notation. In the following we will consider the normalised Gaussian

1

cexp

(−1

2xTAx

)and we define 〈. . .〉 as the average

〈. . .〉 =1

c

∫exp

(−1

2xTAx

). . . dnx.

Here . . . represents an arbitrary expression depending on x that can be averaged. As theGaussian with the prefactor above is normalised we immediately have

〈1〉 = 1.

Gaussian averages of products. We are now interested in averages of the type 〈xk1xk2 . . .〉.A first result about these is the following:

We have〈xk1xk2 . . .〉 = 0

if the number of factors is odd.


This follows immediately from the fact that in case of an odd number of factors the integrandflips sign if we replace x by −x. Hence the contribution of any x to the integral is compensatedby the contribution of −x.

In order to evaluate averages with an even number of factors as well we first determine 〈ejTx〉.This quantity is called the generating function and the parameter j ∈ Rn is called thesource. The advantage of this generating function is that we can generate any average〈xk1xk2 . . .〉 by taking derivatives of the generating function w.r.t. suitable components ofj, and then setting j = 0.

Lemma. ⟨exp

(jTx

)⟩= exp

(1

2jTA−1j

)(3.2)

Proof. The average above can be written as the shifted Gaussian integral⟨exp

(jTx

)⟩= c−1

∫exp

(−1

2xTAx+ jTx

)dnx

We now try to remove the linear term by making a variable transformation the variable trans-formation x = z + A−1j. As this is just a constant shift the Jacobian determinant of thistransformation is 1. We thus obtain

⟨exp

(jTx

)⟩= c−1

∫exp

−1

2(z +A−1j)T︸︷︷︸zT+jTA−1

A(z +A−1j) + jT (z +A−1j)

dnz

where we used that if A is symmetric the same applies to A−1 and hence (A−1)T = A−1. Ifwe expand the brackets we now obtain⟨

exp(jTx

)⟩= c−1

∫exp

(−1

2zTAz −

��1

2jTz −

��1

2zT j − 1

2jTA−1j + �

��jTz + jTA−1j

)dnz

= c−1

∫exp

(−1

2zTAz

)dnz exp

(1

2jTA−1j

)= exp

(1

2jTA−1j

)where we have split off the part of the exponent independent of z and we used (3.1).

Using (3.2) we now consider the case of two prefactors:

Lemma.〈xkxk′〉 = (A−1)kk′

Proof. We can produce two factors xk, xk′ by taking derivatives of the generating functionw.r.t. the corresponding components of the source. Afterwards the remaining exponential canbe turned into 1 by setting j = 0. We thus get

〈xkxk′〉 =∂

∂jk

∂

∂jk′

⟨exp

(jTx

)⟩ ∣∣∣j=0

=∂

∂jk

∂

∂jk′exp

(1

2jTA−1j

) ∣∣∣j=0

.


Now we use the following result for the derivative of quadratic expressions as arising above

∂

∂jk

(1

2jTA−1j

)= (A−1j)k =

∑n

(A−1)knjn.

This can be checked easily using Kronecker deltas and the symmetry of A−1 (actually A−1

could have been replaced by any symmetric matrix). Inserting this result in the precedingformula we obtain

〈xkxk′〉 =∂

∂jk′

∑n

(A−1)knjn exp

(1

2jTA−1j

) ∣∣∣j=0

=∑n

(A−1)knδnk′

= (A−1)kk′

Here we have first taken the derivative w.r.t. jk. Then the derivative w.r.t. jk′ can act on thefactor jn or on the exponential. However the term where it acts on the exponential will vanishbecause it still has the factor jn which is set equal to zero. The remaining steps follow using∂jn∂jk′

= δnk′ .

We now generalise this result to arbitrary products.

Wick’s theorem. Any average 〈xk1xk2xk3 . . .〉 can be obtained by summing over all waysof connecting the factors pairwise by contraction lines as in

〈xk1xk2xk3xk4〉

The contribution of each way of drawing these lines is a product of factors

(A−1)ki,kl

arising from each contraction line connecting xki and xkl .

This means that the average over the contracted factors is evaluated as if the rest of theintegrand were absent.

Example. If we have only two factors there is only one way of connecting the factors pairwiseand we have

〈xk1xk2〉 = 〈xk1xk2〉 = (A−1)k1,k2

as before. For four factors there are three ways of drawing contraction lines

〈xk1xk2xk3xk4〉 = 〈xk1xk2xk3xk4〉+ 〈xk1xk2xk3xk4〉+ 〈xk1xk2xk3xk4〉.

According to Wick’s theorem this leads to

〈xk1xk2xk3xk4〉 = (A−1)k1,k2(A−1)k3,k4 + (A−1)k1,k3(A−1)k2,k4 + (A−1)k1,k4(A−1)k2,k3 .


Proof. The proof generalises the one for products of two factors. The average can be obtainedfrom the generating function via

〈xk1xk2xk3 . . .〉 =∂

∂jk1

∂

∂jk2

∂

∂jk3

. . .⟨exp

(jTx

)⟩ ∣∣∣j=0

=∂

∂jk1

∂

∂jk2

∂

∂jk3

. . . exp

(1

2jTA−1j

) ∣∣∣j=0

.

Now every derivative ∂∂jki

acting on the exponential produces a factor∑n

(A−1)ki,njn

However we later set j = 0. So for any summand n a contribution arises only if the factor jn isremoved by one of the later derivatives ∂

∂jkl. This requires kl = n. As we need this derivative

to remove the factor jn the term where it acts on the exponential is irrelevant. If we have apair of derivatives ∂

∂jki, ∂∂jkl

where one derivative acts on the exponential and the other one

removes a component of j this altogether leads to a factor

(A−1)ki,kl .

We only get a non-vanishing contribution if all derivatives ∂∂jk1

, ∂∂jk2

, ∂∂jk3

, . . . are paired up

in this way. All ways of pairing them up have to be taken into account. A helpful notationfor keeping track of these pairings is to consider the product 〈xk1xk2xk3 . . .〉. Then we drawcontraction lines between xki and xkl if the corresponding derivatives ∂

∂jki, ∂∂jkl

are paired up.

This completes the proof.

Example: One-dimensional integrals. We consider the one-dimensional Gaussian aver-ages with A = 1,

〈. . .〉 =

∫ ∞−∞

√2πe−

12x2. . . dx .

We obtain 〈x〉 = 0, 〈x3〉 = 0, etc because the integrand is odd in x or, equivalently, because anodd number of factors can’t be grouped into pairs so there are no permissible ways of drawingcontraction lines. Furthermore we have, following on from our earlier example,

〈x2〉 = 〈xx〉 = 1

and

〈x4〉 = 〈xxxx〉+ 〈xxxx〉+ 〈xxxx〉 = 3.

In general for 〈x2n〉 we start with a contraction involving the first factor x. There are 2n− 1choices for the factor it is contracted with. Afterwards 2n − 2 uncontracted factors are left.If we draw a contraction line involving the first of these factors we have 2n − 3 choices forthe factor it is contracted with. Due to A = 1 each way of drawing contraction lines justcontributes 1 to the result. Continuing like this we obtain

〈x2n〉 = (2n− 1)(2n− 3)(2n− 5) . . .

which is often denoted by the double factorial

〈x2n〉 = (2n− 1)!! .


For example we have

〈x6〉 = 5 · 3 · 1 = 15

and

〈x8〉 = 7 · 5 · 3 · 1 = 105.

Wick’s theorem for Fresnel integrals. We now consider Fresnel integrals, with e−12xTAx

replaced by

ei2xTAx. (3.3)

We again require A to be symmetric. There is no longer a reason to require A to be positivedefinite, but we still assume it has no vanishing eigenvalues and is therefore invertible. We willsee that all that changes in this case is an extra factor i in Wick’s theorem.

To show this we note that the one dimensional Gauss integral is turned into the one dimensionalFresnel integral seen in section 2.1,

∫ ∞−∞

ei2ax2

=

√2π

|a|eiπ4

sgn a

The analogous formula in n dimensions is∫Rnei2xTAxdnx =

(2π)n/2√|detA|

eiνπ4 ≡ c

where ν is the difference between the number of positive eigenvalues of A and the number ofnegative eigenvalues. ν is thus the appropriate generalisation of sgn a. We are now interestedin the average

〈. . .〉 =1

c

∫ei2xTAx . . . .

As before the averages of products 〈xk1xk2 . . .〉 can be obtained by taking derivatives of asuitably defined generating function. As in (3.3) we flipped the signs and inserted a factor icompared to the Gaussian case, we make the same change in the generating function and write

〈e−ijTx〉 = e−

i2jTA−1j .

We then obtain

〈xkxk′〉 = − ∂

∂jk

∂

∂jk′〈e−ij

Tx〉∣∣j=0

= − ∂

∂jk

∂

∂jk′e−

i2jTA−1j

∣∣j=0

and the same calculation as for the Gauss integral leads to

〈xkxk′〉 = i(A−1)kk′ .

An analogous factor arises in Wick’s theorem as the contribution of each contraction. This

is the change one would have expected naively by realising that ei2xTAx is obtained from the

previous Gaussian via the replacement A→ −iA which implies A−1 → iA−1.


3.3 Anharmonic oscillator

We now apply Wick’s theorem to path integrals. As an example we consider the anharmonic orperturbed harmonic oscillator. This is just the harmonic oscillator with a small perturbationεx4 (with ε� 1) added to the potential, leading to the Lagrangian

L = T − U =1

2mx2 − 1

2mω2x2 − εx4.

Our aim will be to evaluate the path integral for the propagator K(0, 0, t) describing motionfrom x0 = 0 to xf = 0, and to extract the ground state energy from this result using theprocedure from section 2.5. The main technical challenge will be to go from the versionof Wick’s theorem with discrete indices to a situation where time appears as a continuousparameter. This will be done by making the ’natural’ replacements, with sums replaced byintegrals, vectors by functions, matrices by operators. (These replacements could be justifiedformally but we will sometimes appeal to intuition about what is the natural continuousgeneralisation of a discrete object.) The path integral for the anharmonic oscillator is

K(0, 0, t) =

∫x(0)=x(t)=0

D[x] exp

(i

~

∫ t

0

[1

2mx(t′)2 − 1

2mω2x(t′)2 − εx(t′)4

]dt′).

Comparison to the discrete case. Here the function x(t′) is the continuous analogue ofthe vector x = (x1, x2, . . .) considered earlier. The analogue of the quadratic term i

2xTAx

involves the quadratic parts of the action, associated to the harmonic oscillator. They can bewritten as

i

~S0 =

i

~

∫ t

0

[1

2mx(t′)2 − 1

2mω2x(t′)2

]dt′

where S0 indicates the action of the harmonic oscillator. The integral here is obviouslyquadratic but it would be closer to i

2xTAx if we could write the integrand with x(t′) on

the left and on the right and an operator in between. To bring it into this form we integrateby parts, ∫ t

0x(t′)2dt′ = x(t′)x(t′)

∣∣t0︸︷︷︸

=0

−∫ t

0x(t′)x(t′)dt′ = −

∫ t

0x(t′)

∂2

∂t′2x(t′)dt′

leading to

i

~S0 =

i

2

∫ t

0x(t′)

(−m

~

)( ∂2

∂t′2+ ω2

)x(t′).

Hence the operator

A =(−m

~

)( ∂2

∂t′2+ ω2

)

is the analogue of the matrix A appearing in the discrete version of Wick’s theorem. Thisreplacement is natural as differential (and other) operators act on functions in a similar wayas matrices act on vectors.

3.3. ANHARMONIC OSCILLATOR 27

Inverse of A. For Wick’s theorem we need the inverse of A which should be an operatoras well. We now assume that A−1 is an operator whose application to a function x(t′) can bewritten in the form

(A−1x)(t′) =

∫ t

0G(t′, t′′)x(t′′)dt′′ (3.4)

where the parameters t′, t′′ take a role similar to indices of matrices an vectors. G(t′, t′′) iscalled the integral kernel of A−1. To determine A−1 and G(t′, t′′) we use that AA−1 has to bethe identity. This implies that for all functions x(t′) we must have

x(t′) = AA−1x(t′) = A

∫ t

0G(t′, t′′)x(t′′)dt′′ =

∫ t

0AG(t′, t′′)x(t′′)dt′′.

Now comparing the first and the final expressions in this line we see that G(t′, t′′) has to satisfy

AG(t′, t′′) = δ(t′ − t′′)

which is the differential equation(−m

~

)( ∂2

∂t′2+ ω2

)G(t′, t′′) = δ(t′ − t′′). (3.5)

In addition, when considering operators acting on functions it is important to consider theprecise space of functions they are acting on. In the present problems we are interested infunctions x(t′) that satisfy x(0) = x(t) = 0. Hence application of A according to (3.4) to afunction satisfying these requirements should return another function satisfying them. Thisleads to the conditions

G(0, t′′) = G(t, t′′) = 0 (3.6)

guaranteeing that∫ t

0 G(t′, t′′)x(t′′)dt′′ vanishes for t′ = 0 or t′ = t. We could now use (3.5) and(3.6) to determine G(t′, t′′) by calculation but it will be slightly easier to just state the resultand prove that it’s true.

Proposition. The solution G(t′, t′′) subject to the requirements (3.5) and (3.6) is

G(t′, t′′) = − ~mω sinωt

{sinωt′ sinω(t′′ − t) for t′ ≤ t′′

sinωt′′ sinω(t′ − t) for t′ > t′′.

Note that in spite of the case distinction G(t′, t′′) is continuous, and the lower formula wouldgive the same result for G(t′, t′) as the upper one. However crucially we will see that the firstderivative makes a jump at t′ = t′′.

Proof. The boundary conditions (3.6) are satisfied because for t′ = 0 we are in the first caseand the vanishes result vanishes due to sinωt′ = 0. For t′ = t we are in the second case andthe result vanishes due to sinω(t′ − t) = 0. For t′ 6= t′′ the differential equation (3.5) simplyboils down to the oscillator equation(

∂2

∂t′2+ ω2

)G(t′, t′′) = 0.


which is satisfied due to the terms sinωt′ and sinω(t′ − t). The tricky part is to show that in(3.5) the delta function arises for t′ = t′′. To show this we evaluate the first derivative

∂

∂t′G(t′, t′′) = − ~ω

mω sinωt

{cosωt′ sinω(t′′ − t) for t′ < t′′

sinωt′′ cosω(t′ − t) for t′ > t′′.

Crucially this result has a jump at t′ = t′′. Now using the Heaviside function

Θ(x) =

{1 for x > 0

0 for x < 0

we can rewrite our result as

∂

∂t′G(t′, t′′) = − ~ω

mω sinωt[cosωt′ sinω(t′′ − t)

+Θ(t′ − t′′)(sinωt′′ cosω(t′ − t)− cosωt′ sinω(t′′ − t))].

If we then apply ∂∂t′ for a second time, the application to the trigonometric functions will only

give terms that we already know will be compensated by ω2G(t′, t′′). However in addition tothese we have to take into account the term where ∂

∂t′ acts on Θ(t′− t′′). To evaluate this termwe use that the derivative of the Heaviside function is the delta function. We thus obtain(

∂2

∂t′2+ ω2

)G(t′, t′′) = − ~ω

mω sinωtδ(t′ − t′′)(sinωt′′ cosω(t′ − t)− cosωt′ sinω(t′′ − t))

Now in the factor following the delta function we can replace t′′ by t′ as this factor onlycontributes if the argument of the delta function is zero. Then we can use a trigonometricidentity to simplify

sinωt′ cosω(t′ − t)− cosωt′ sinω(t′ − t) = sin(ωt′ − ω(t′ − t)) = sinωt

and we obtain (∂2

∂t′2+ ω2

)G(t′, t′′) = − ~

mδ(t′ − t′′)

which implies (3.5).

Propagator. We now return to the path integral for the propagator. For the anharmonicoscillator we have

Kanh(0, 0, t) =

∫x(0)=x(t)=0

D[x] exp

(i

~

∫ t

0

[1

2mx(t′)2 − 1

2mω2x(t′)2 − εx(t′)4

]dt′)

=

∫x(0)=x(t)=0

D[x] exp

(i

~

∫ t

0

[1

2mx(t′)2 − 1

2mω2x(t′)2

]dt′)

exp

(− iε~

∫ t

0x(t′)4dt′

).

We first have to translate this into our notation with averages 〈. . .〉. We have

〈. . .〉 =1

c

∫x(0)=x(t)=0

D[x] exp

(i

~

∫ t

0

[1

2mx(t′)2 − 1

2mω2x(t′)2

]dt′). . .

where c is the integral involving only the quadratic terms. In the present case this is just thepropagator of the harmonic oscillator,

c = Kharm(0, 0, t) =

∫x(0)=x(t)=0

D[x] exp

(i

~

∫ t

0

[1

2mx(t′)2 − 1

2mω2x(t′)2

]dt′).


In this notation the propagator of the anharmonic oscillator can be written as

Kanh(0, 0, t) = Kharm(0, 0, t)

⟨exp

(− iε~

∫ t

0x(t′)4dt′

)⟩. (3.7)

As ε is small we can expand the exponential in a Taylor series. If we go up to quadratic orderwe obtain

Kanh(0, 0, t) = Kharm(0, 0, t)

(1− iε

~

∫ t

0dt′〈x(t′)4〉

+1

2

(− iε~

)2 ∫ t

0dt′∫ t

0dt′′〈x(t′)4x(t′′)4〉+O(ε3)

)

where we have interchanged the time integral and the average, and we renamed one of theintegration variables in the squared integral into t′′.

We now consider the term linear in ε, and use Wick’s theorem to evaluate the average 〈x(t′)4〉involved in it. The factors x(t′) here are the continuous analogues of the factors xk we had inthe discrete formulation of Wick’s theorem. If we contract the x(t′)’s as we did with the xk’swe obtain

〈x(t′)4〉 = 〈x(t′)x(t′)x(t′)x(t′)〉+ 〈x(t′)x(t′)x(t′)x(t′)〉+ 〈x(t′)x(t′)x(t′)x(t′)〉.

Now in the discrete case (with Fresnel integrals) every contraction line between xki and xkl gavea contribution i(A−1)ki,kl . In the continuous case this means that contractions between x(t′)and x(t′′) should lead to give iA−1 evaluated at the times t′ and t′′. The obvious interpretationof this is to take the kernel of A−1 and write iG(t′, t′′). In the formula above all contractionshence give factors iG(t′, t′) and we obtain

〈x(t′)4〉 = 3(iG(t′, t′))2.

With

G(t′, t′) = −~ sinωt′ sinω(t′ − t)mω sinωt

. (3.8)

we thus obtain

Kanh(0, 0, t) = Kharm(0, 0, t)

(1 +

3iε

~

∫ t

0G(t′, t′)2dt′ +O(ε2)

)(3.9)

where we have neglected all higher-order terms, including the quadratic terms still writtenabove.

Ground state energy. We now use our previous result to study how the ground state energyof the harmonic oscillator is changed by the perturbation. Hence we replace i

~ t→ β, i~ t′ → β′

as in section 2.5; this leads to 〈0|e−βHanh |0〉 which should be proportional to e−βE0 in the limitβ →∞. (We stress that here 0 indicates the initial and final position at the origin, this shouldnot be confused with the notation |0〉 sometimes used for the ground state.) Approximating

〈0|e−βHanh |0〉 in our limit thus allows to read off the ground state energy E0.


The replacement i~ t→ β leads to

〈0|e−βHanh |0〉 = 〈0|e−βHharm |0〉

(1 + 3ε

∫ β

0G

(~iβ′,

~iβ′)2

dβ′ +O(ε2)

)(3.10)

where we have

G

(~iβ′,

~iβ′)

= −~ sin

(ω ~i β′) sin

(ω ~i (β′ − β)

)mω sin

(ω ~i β) . (3.11)

For large β we can approximate

sin

(ω~iβ

)=

1

2i

(eω~β − e−ω~β

)≈ 1

2ieω~β.

and analogously we obtain for large β′

sin

(ω~iβ′)≈ 1

2ieω~β

′.

Note that we have 0 ≤ β′ ≤ β so β′ is not large in the beginning of the integration intervalbut it is large for more and more of the integration interval the larger β gets, so substitutingthe above approximation in the integral indeed gives a good approximation. With

− sin

(ω~i(β′ − β)

)= sin

(ω~i(β − β′)

)≈ 1

2ieω~(β−β′)

we obtain

G

(~iβ′,

~iβ′)≈ ~

2imω

which means that for large β all exponentials in the integrand cancel. Substitution into (3.10)and integration now yields the following result for large β

〈0|e−βHanh |0〉 ≈ 〈0|e−βHharm |0〉

(1− 3

4ε

(~mω

)2

β +O(ε2)

).

We now use the result from section 2.5 for the first factor, and we write the second factor asan exponential using that ε is small. This gives the following proportionality for small ε andlarge β

〈0|e−βHanh |0〉 ∝ e−β12~ωe−

34ε( ~mω )

2β = e

−β[

12~ω+ 3

4ε( ~mω )

2].

We thus conclude that, considering only the linear order in ε, the perturbation changes theground state energy into

E0 =1

2~ω +

3

4ε

(~mω

)2

.

Feynman diagrams. We now introduce a helpful notation for writing down contractionslike the ones we considered for 〈x(t′)4〉. We also allow for contractions between x with differenttime arguments. In the new notation any contraction line

x(t′)x(t′′)


is indicated by a line connecting vertices that represent t′ and t′′. According to Wick’s theoremthe contribution of this line is iG(t′, t′′). Each time argument appearing in the problem isdenoted only by a single vertex, hence a contraction line involving identical time argumentslike

x(t′)x(t′)

is represented by a loop starting and ending at the same vertex t′. The usual convention alsorequires that any time vertex that has more than one end of a line attached is automaticallyintegrated over. With this convention each of the three ways of contracting x(t′) in∫ t

0dt′〈x(t′)4〉

is denoted by

As there are three choices of contractions one says that this diagram has multiplicity 3 and∫ t0 dt

′〈x(t′)4〉 yields 3 times the diagram depicted.

ε2 contribution. To illustrate these diagrams further we also consider the ε2 contributionto K(0, 0, t) which is proportional to∫ t

0dt′∫ t

0dt′′〈x(t′)4x(t′′)4〉. (3.12)

The associated diagrams are the following:

• First we consider the case that all contraction lines connect x with the same argument,e.g.

∫ t

0dt′∫ t

0dt′′〈x(t′)x(t′)x(t′)x(t′)x(t′′)x(t′′)x(t′′)x(t′′)〉

=

∫ t

0dt′∫ t

0dt′′(iG(t′, t′))2(iG(t′′, t′′))2

There are 9 contributions of this type as there are 3 choices for contracting the x(t′)among themselves and 3 choices for contracting the x(t′′) among themselves. The overallcontribution to (3.12) can thus be written as


• Next we assume that there are two contractions between x(t′) and x(t′′) and one con-traction each among the x(t′) and among the x(t′′), as for example in

∫ t

0dt′∫ t

0dt′′〈x(t′)x(t′)x(t′)x(t′)x(t′′)x(t′′)x(t′′)x(t′′)〉

=

∫ t

0dt′∫ t

0dt′′(iG(t′, t′′))2iG(t′, t′)iG(t′′, t′′)

The corresponding diagram has multiplicity 72. To understand this note that there are(42

)= 6 ways of grouping the four factors x(t′) into two factors to be contracted with

each other and two factors to be contracted with x(t′′). An analogous 6 choices arisefrom grouping the x(t′′) in a similar way. Then there are two ways in which we can drawthe required contractions lines between x(t′) and x(t′′). The contribution to (3.12) istherefore

• Finally all contractions could be between x(t′) and x(t′′). In this case the first x(t′) canbe contracted with four possible x(t′′), for the next x(t′) there are three x(t′′) left, andthen there are two and finally only one. The multiplicity is thus 4! = 24 and we obtain

The multiplicities sum to 105 = 7!! which is the overall number of possibilities to contract eightfactors pairwise.

Performing the integrals we could now determine the ε2 contribution to K(0, 0, t). This wouldgive an even better description of wave propagation for the anharmonic oscillator than (3.9).

In addition after the substitution i~ t → β we can get 〈0|e−βH |0〉 up to order ε2 which allows

to determine E0 up to order ε2. However these calculations will be omitted here.

Chapter 4

Second quantisation

We now want to introduce the formalism of second quantisation, which provides a way to dealwith systems involving multiple particles.

4.1 Two particles

We start by considering systems with two particles. A possible Hamiltonian for a two-particlesystem is

H =p2

1

2m+ U(r1) +

p22

2m+ U(r2) + Uint(r1, r2).

Here r1 and r2 are the positions of the two particles, p1 and p2 are the corresponding mo-mentum operators, m is the mass (assumed to be the same for both particles) and U is thepotential (also assumed to be the same). In addition there is a potential term Uint(r1, r2)that depends on the positions of both particles and describes their interaction. For example,electric attraction/repulsion or gravitational attraction leads to Uint(r1, r2) = a

|r1−r2| withsuitably chosen a.

Hilbert space. The Hilbert space of the two-particle system is the space of all possible wave-functions ψ(r1, r2) depending on both positions. As usual these wavefunctions are assumed tobe square integrable.

Basis. Now assume that we are using a basis ψi(r) for the Hilbert space of single-particle

wavefunctions. (E.g. these could be eigenfunctions of the single particle Hamiltonian p2

2m+U(r)but we could also use any other basis.) In Dirac notation these basis functions will be writtenas |i〉. Then a suitable basis for the Hilbert space of two-particle wavefunctions is providedby the products of single-particle basis functions ψi(r1)ψj(r2). In Dirac notation these basisfunctions are indicated by the tensor products |i〉⊗ |j〉, or if one omits the tensor product signone simply writes |i〉|j〉.

Indistinguishable particles. Importantly, quantum mechanical particles of the same type(such as electrons, protons, etc) are indistinguishable from each other. This means that fora system of, say, two such particles the result of a measurement might indicate that oneparticle is at position r1 and the other particle is at position r2. However as the particles are

33

34 CHAPTER 4. SECOND QUANTISATION

indistinguishable we can’t tell apart the case that the first particle is at r1 and the second oneis at r2 and the case that the first one is at r2 and the second one is at r1.

This has important consequences for the corresponding wavefunctions. The probability densityfor finding the first particle at r1 and the second particle at r2 is given by |ψ(r1, r2)|2 whereasthe probability density for finding the first particle at r2 and the second particle at r1 is|ψ(r2, r1)|2. If both events are indistinguishable we must have

|ψ(r2, r1)|2 = |ψ(r1, r2)|2

and hence

ψ(r2, r1) = eiφψ(r1, r2).

It turns out (this is a law of nature) that the choice of eiφ depends on the type of particle:

• For bosons (e.g. photons - the particles associated with light) we always have eiφ = 1.

• For fermions (e.g. electrons, protons,. . . ) we always have eiφ = −1.

• Other choices of eiφ are not possible for fundamental particles.

Particles composed of these building blocks, e.g., atoms or molecules may be bosons or fermions.It turns out that more elaborate ways of building ’effective’ particles from fundamental buildingblocks even allow for particles with other eiφ, so-called anyons.

Spin. Quantum mechanical particles appearing in nature also often have spin. In spin sys-tems the wavefunction assumes a vector form with components ψσ(r). Equivalently we couldregard the index σ as part of the argument and write ψ(σ, r) if we remember that σ has tobe discrete. For systems with spin everything said here carries over if we replace r by (σ, r).Writing r only is just a convenient way to simplify notation.

If one goes deeper in the theory of spin one realises that for bosons the number of spin com-ponents must be odd (this includes the case of single component wavefunctions and hence nospin) whereas for fermions it must be even.

Hilbert space. The Hilbert space for systems of two bosons is the space of (square-integrable)wavefunctions ψ(r1, r2) satisfying ψ(r2, r1) = ψ(r1, r2). For fermions the condition is replacedby ψ(r2, r1) = −ψ(r1, r2)

Basis. The basis functions of this new Hilbert space also have to be (anti-)symmetric withrespect to particle exchange. Using the basis of single particle wavefunctions ψi(r) (anti-)symmetric functions can be constructed as follows:

ψij(r1, r2) = C(ψi(r1)ψj(r2)± ψj(r1)ψi(r2)). (4.1)

Here the plus sign is for bosons and the minus sign for fermions, and C is a normalisationconstant to be determined later. ψij(r1, r2) is the only bosonic/fermionic state with one particle

4.2. N PARTICLES 35

in the single-particle state i and one article in the single particle state j. This contrasts withthe case of distinguishable particles where we have (for i 6= j) two basis functions with oneparticle in state i and one in state j, namely ψi(r1)ψj(r2) and ψj(r1)ψi(r2). In Dirac notationour bosonic/fermionic basis states can be written as

|i, j〉 = C(|i〉|j〉 ± |j〉|i〉).

Pauli exclusion principle. Importantly, for fermions we have ψii(r1, r2) = 0, i.e., there isno basis state where both particles are in the single-particle state i. The inability of fermionsto be both in the same state is known as Pauli’s exclusion principle. (Note that if spin is takeninto account, this means that the two particles cannot be in the same state as characterisedby position and spin!)

Normalisation. We still have to determine C. We demand our single-particle states to benormalised and orthogonal to each other, i.e.,∫

dnr ψ∗i (r)ψi′(r) = δii′ .

Then the same applies to the two-particle states ψi(r1)ψj(r2) where indistinguishability is notyet taken into account, ∫

dnr1

∫dnr2 (ψi(r1)ψj(r2))∗(ψi′(r1)ψj′(r2))

=

(∫dnr1ψi(r1)∗ψi′(r1)

)(∫dnr2ψj(r2)∗ψj′(r2)

)= δii′δjj′ (4.2)

We now want to choose C such that the ψij(r1, r2) defined in (4.1) satisfies

1 =

∫dnr1d

nr2|ψij(r1, r2)|2.

We first consider i 6= j, and assume that C is positive. If we insert (4.1) we then get C2 timesthe norm of ψi(r1)ψj(r2) which is 1, C2 times the norm of ψj(r1)ψi(r2) which is also 1, C2

times scalar products of ψi(r1)ψj(r2) and ψj(r1)ψi(r2) which are zero. Hence we have

1 = C2(1 + 1)⇒ C =1√2.

For bosonic states with two particles occupying the single particle state i we have

ψii(r1, r2) = ψi(r1)ψi(r2)

which is normalised due to (4.2).

4.2 N particles

Hilbert space. For N -particle systems the wavefunction can be written as ψ(r1, r2, . . . , rN ).The conditions on ψ are analogous to the case of two particles. For bosons the wavefunctionhas to be symmetric w.r.t. exchanges of any two of the arguments, and for fermions it has tobe antisymmetric w.r.t. such exchanges.


Basis. We want to build basis states similar to the case of two particles, first for bosons. Letψi1,...,ıN (r1, . . . , rN ) be the many particle state in which the single-particle states i1, . . . , iN areoccupied. (For bosons some or all of the states may coincide. ) One might at first consider theproduct state ψi1(r1) . . . ψiN (rN ). However this is not symmetric w.r.t. exchanging argumentsso in analogy to the case of two particles we symmetrise and consider

ψi1,...,ıN (r1, . . . , rN )

= C(ψi1(r1) . . . ψiN (rN ) + all terms where indices are interchanged).

Here one could exchange either the indices of the ψ’s or the indices of the r’s but the usualconvention is to exchange the former.

For fermions any exchange of two particles should flip the sign. Hence for the terms withexchanged indices we imagine the exchanges to be performed in steps, and in each step justtwo indices are interchanged. If the number of these steps (transpositions) is even the signflips compensate. For the terms involving an odd number of transpositions there is an overallminus sign.

Permutations. To formalise these results mathematically we need permutations:

• A permutation of the integers 1, 2, . . . , N is defined as a bijective mapping between thesenumbers. As a consequence of the mapping being bijective no two numbers may bemapped to the same number.

• One can show that every permutation may be obtained by successive application oftranspositions. These are permutations in which just two numbers interchanged, and allother numbers are mapped to themselves.

• A given permutation can be written in terms of different numbers of transpositions. (E.g.one could always add another transposition that interchanges two numbers, and a furthertransposition that undoes this exchange.) However one can show for a given permutationthe number of transpositions is always odd or even. Hence it makes sense to define thesign of a permutation π as

signπ =

{+1 if the number of transpositions is even

−1 if the number of transpositions is odd.

With permutations the basis functions introduced above can be written as

ψi1,...,iN (r1, . . . , rN ) = C∑π

ψiπ(1)(r1) . . . ψiπ(N)

(rN )×

{1 for bosons

signπ for fermions.

We note that using the representation of the determinant in terms of a sum over permuta-tions the result for the fermionic case can be written as a determinant (the so called Slaterdeterminant).

Normalisation. Similarly as for two-particle states we have to choose C such that the basisfunctions are normalised. If one works this out (this will be done in an exercise for the caseN = 3) one obtains for fermions

C =1√N !.

4.3. FOCK SPACE 37

For bosons we also have to take into account the possibility that there are several particles inthe same state. Then one obtains

C =1√

N !∏i ni!

where ni is the number of particles in state i.

4.3 Fock space

In quantum mechanics we can have the situation that we don’t know how many particlesthere are in our system. The number of particles is an observable (measurable quantity) likepositions and momenta. If we have not performed a measurement of the particle number (orset the system up to have a given number of particles from the start) the number of particlesmay be unknown similar to a position or momentum being unknown if we have not made acorresponding measurement. States where the number of particles is not fixed can be writtenas quantum superpositions of states with different numbers of particles.

In this superposition we even have to allow for the possibility of having no particles at all. Thequantum state with no particles at all is called the vacuum. In Dirac notation the vacuumis denoted by |0〉. We stress that this is a valid quantum mechanical state and it has to bedistinguished from zero.

If we denote by FN the Hilbert space of (bosonic or fermionic) N particle wavefunctions, theHilbert space of all permissible states (the Fock space) is formed by linear combinations ofelements of all FN . The space of all linear combinations of elements of a number of vectorspaces is called the direct sum of these vector spaces. Hence the Fock space can be formallydefined as

F =

∞⊕N=0

FN

where⊕

indicates a direct sum.

Note: If we have a linear combination of states with different N its component for each Ncan be written as a wavefunction with N arguments. But the linear combination as a wholecannot be written as a wavefunction with a well-defined number of parameters.

4.4 Creation and annihilation operators

Occupation number representation. In order to do calculations in Fock space it is helpfulto use creation and annihilation operators. To define these operators we represent our statesin occupation number representation. In this representation the multiple-particle basis statesare indicated by |n1, n2, . . .〉 where ni indicates the number of particles that are in the singleparticle state i. Translated into the notation used earlier we thus have

|n1, n2, . . .〉 = | 1, . . . , 1︸︷︷︸n1 entries

, 2, . . . , 2︸︷︷︸n2 entries

, . . .〉.

The occupation number representation can also be used for fermions; in this case all ni areeither 0 or 1. The bra associated to |n1, n2, . . .〉 is written as

〈n1, n2, . . . |.


Creation and annihilation operators for bosons. The creation operator a†i is definedsuch that it increases the number of particles in state i by 1 and multiplies the state with afactor

√ni + 1:

a†i |n1, . . . , ni, . . .〉 =√ni + 1|n1, . . . , ni + 1, . . .〉. (4.3)

The annihilation operator ai removes a particle in state i and multiplies with√ni:

ai|n1, . . . , ni, . . .〉 =√ni|n1, . . . , ni − 1, . . .〉. (4.4)

In these definitions the square root factors are included for the following reasons: They makethe creation and annihilation operators analogous to the raising and lowering operators arisingin the algebraic approach to the harmonic oscillator. Moreover, if the annihilation operator aiis applied to a state with ni = 0 the prefactor makes sure that the resulting state is identicallyzero. This is needed for consistency as we can’t have a multiple-particle states with −1 particlesin state i. Finally, some of our later results (e.g. for the particle number operator) look muchnicer with the present definitions.

Adjointness. We still have to show that the ai and a†i thus defined are adjoints of each

other. To do so we consider the following matrix elements of a†i ,

〈n1, n2, . . . , ni, . . . |a†i |n1, n2, . . . , ni, . . .〉= 〈n1, n2, . . . , ni, . . . |

√ni + 1|n1, n2, . . . , ni + 1, . . .〉

=√ni + 1 δn1,n1δn2,n2 . . . δni,ni+1. (4.5)

Now we know that this expression has to be complex conjugated if we taken the adjoint of allexpressions involved (exchanging the bra and the ket) and invert their ordering. If ai and a†iare mutually adjoint the above expression thus has to be complex conjugate to

〈n1, n2, . . . , ni, . . . |ai|n1, n2, . . . , ni, . . .〉= 〈n1, n2, . . . , ni, . . . |

√ni|n1, n2, . . . , ni − 1, . . .〉

=√ni δn1,n1δn2,n2 . . . δni,ni−1. (4.6)

This is indeed correct. Both expressions are nonzero only if ni = ni+1 and all other occupationnumbers coincide. In this case the square roots also coincide, and the expressions are real socomplex conjugation does not change anything.

For a full proof of adjointness we still have to show that an analagous result holds for arbitrarybra and ket states. This will be worked out in an exercise.

Commutators. The commutators [A,B] = AB − BA of the creation and annihilation op-erators are

[ai, aj ] = 0, [a†i , a†j ] = 0, [ai, a

†j ] = δij . (4.7)

Proof: Each creation and annihilation operator just modifies the occupation number corre-sponding to its index, and multiplies with a factor only depending on this number. Hence the

4.5. HAMILTONIAN IN SECOND QUANTISATION 39

ordering or creation and annihilation operators with different indices does not matter, and alloperators with i 6= j commute.

For i = j we trivially have [ai, ai] = 0 and [a†i , a†i ] = 0. The only nontrivial commutator is

[ai, a†i ]. If we apply this commutator to a state | . . . , ni, . . .〉 we obtain

[ai, a†i ]| . . . , ni, . . .〉

= aia†i | . . . , ni, . . .〉 − a

†iai| . . . , ni, . . .〉

= ai√ni + 1| . . . , ni + 1, . . .〉 − a†i

√ni| . . . , ni − 1, . . .〉

= (ni + 1)| . . . , ni, . . .〉 − ni| . . . , ni, . . .〉= | . . . , ni, . . .〉 (4.8)

Here we used only the definitions (4.3) and (4.4). Note that in the next-to-last line we appliedai to a state where the occupation number ni is replaced by ni + 1, hence the factor

√ni in

(4.4) is replaced by√ni + 1 which has to be multiplied with the

√ni + 1 already there; the

second term in that line is obtained similarly. As [ai, a†i ] acts like 1 on all basis states, it has

to act like 1 on all states of the Fock space.

Particle number operator. If we consider the second terms in all lines of (4.8) we see that

a†iai| . . . , ni, . . .〉 = ni| . . . , ni, . . .〉

i.e. a†iai is the operator giving the number of particles in state i.

4.5 Hamiltonian in second quantisation

We now want to write down the Hamiltonian of a multiple-particle quantum system in secondquantisation, using creation and annihilation operators.

Example. We start by considering an example system where the particles can only be ondiscrete sites, labelled by 1, 2, 3, . . .. For this system the single-particle wavefunctions are justvectors ψ(1)

ψ(2)...

The single-particle Hamiltonian is chosen to mimick

Hψ(x) =

(− ~2

2m

d2

dx2+ U(x)

)ψ(x).

The discrete analogue of the second derivative of ψ(x) is1

ψ(x+ 1)− 2ψ(x) + ψ(x− 1)

1To show this, note that the first derivative is given by ψ′(x) = lim∆x→0ψ(x+∆)−ψ(x)

∆x. In the discrete case

we can’t take ∆x to zero, the best thing we can do is take ∆x to 1. This gives ψ(x+1)−ψ(x) as the analogue ofthe first derivative, but we could also take ψ(x)−ψ(x−1). If we take f(x) = ψ(x+1)−ψ(x) as the counterpartof the first derivative and g(x) = f(x)− f(x− 1) as the counterpart of the second derivative we get

g(x) = f(x)− f(x− 1) = (ψ(x+ 1)− ψ(x))− (ψ(x)− ψ(x− 1)) = ψ(x+ 1)− 2ψ(x) + ψ(x− 1)

as used above.


Hence for the discrete system we have

(Hψ)(x) = − ~2

2m(ψ(x+ 1)− 2ψ(x) + ψ(x− 1)) + U(x)ψ(x). (4.9)

In matrix form the Hamiltonian thus becomes

H = − ~2

2m

−2 11 −2 1

1 −2. . .

. . .. . .

+

U(1)

U(2)U(3)

. . .

(4.10)

Multi-particle Hamiltonian. Now let us consider a system with discrete sites, but al-low for multiple particles. Then the corresponding Hamiltonian will act on wavefunctionsψ(x1, x2, . . . , xN ). It is given by

Hmult =

N∑p=1

Hp +1

2

∑p,p′=1,...,N

p6=p′

Uint(xp, x′p). (4.11)

Here the part Hp gives the energy of the p-th particle. It is defined as in (4.9) w.r.t. thevariable xp taking the role of x. In addition different particles can interact. For example anelectric or gravitational interaction between the particles p and p′ would lead to an interactionpotential of the form a

|xp−xp′ |. In general the potential due to interaction of particles at xp and

xp′ is denoted by Uint(xp, xp′). This needs to be summed over all particles p and p′. However,in the sum we have to exclude the terms p = p′ as a particle does not interact with itself. (Forexample, a particle does not gravitationally attract or electrically repel itself.) Moreover, ifwe let the sums over p and p′ run over all particles the interaction between two particles willformally be counted twice as we could choose the first particle as p and the second one as p′

or the other way around. To compensate this we have divided by 2 in (4.11).

Second quantisation. We now want to write Hmult in the formalism of second quantisation,using creation and annihilation operators a†i , ai. To do so we have to choose a basis of singleparticle wavefunctions ψi(x). We simply take the wavefunctions localised at the site i, i.e.ψi(x) = δxi. (Note that here x is a discrete argument.)

We have to consider all terms present in (4.10) and (4.11):

• Potential: Each particle at site i feels the potential U(i), and the number of all particles

at site i can be obtained by applying the particle number operator a†iai. Hence the totalpotential is obtained by applying the operator∑

i

U(i)a†iai. (4.12)

• Single particle Hamiltonian: For the total contribution of the single particle Hamil-tonians we will only make an educated guess (but one can prove that it’s correct). Weare guided by the observation that the potential is represented by a diagonal matrixin (4.10), and this gives rise to a sum over the diagonal elements multiplied with a†iaiwhere i is the index of the corresponding row as well as the corresponding column. By

4.5. HAMILTONIAN IN SECOND QUANTISATION 41

contrast the full single particle Hamiltonian in (4.9) is given by a matrix containing off-diagonal elements. The natural generalisation of the formula (4.12) to the off-diagonalsingle-particle Hamiltonians as in (4.10) is therefore∑

i,j

H(i, j)a†iaj .

Here H(i, j) are the matrix elements.

• Interactions: The interaction potential between two particles at sites i, j can be writtenas Uint(i, j). If i 6= j such a potential arises for all pairs of particles at i and j. To getthe number of pairs we multiply the number of particles in state i, obtained using theoperator a†iai, with the number of particles in state j, obtained using the operator a†jaj .In the case i = j both particles are at the same site, so they could potentially be thesame. However we should not include self-interaction. Hence in the case i = j the numberof choices for the second particle should be reduced by 1, meaning that in general thenumber of choices for the second particle is a†jaj − δij . The overall interaction potentialis therefore given by the operator

1

2

∑i,j

Uint(i, j)a†iai(a

†jaj − δij).

Here the reasons for the factor 12 are the same as in Eq. (4.11).

Theorem: The combination of operators above can be simplified as

a†iai(a†jaj − δij) = a†ia

†jajai.

Proof: We have

a†ia†jajai = a†ia

†jaiaj = a†i (aia

†j − (aia

†j − a

†jai)︸︷︷︸

=[ai,a†j ]=δij

)aj = a†iaia†jaj − δija

†iaj

= a†iai(a†jaj − δij)

Here we have used in the first step that [aj , ai] = 0, and in the final step that δija†iaj

only contributes if i = j.

With these results the full multiple-particle Hamiltonian can be written as

Hmult =∑i,j

H(i, j)a†iaj +1

2

∑i,j

Uint(i, j)a†ia†jajai.

Continuous case. We also want to state the result for the case of continuous positions inRn instead of discrete sites. In this case the indices i, j are replaced by r, r′ ∈ Rn. Theoperator a†(r) creates a particle at position r. In this case the potential term (4.12) becomes∫dnr U(r)a†(r)a(r) =

∫dnr a†(r)U(r)a(r). The off-diagonal parts in (4.10) were arising from

a discretisation of the second derivative in the kinetic energy− ~2

2m∇2 (where for n = 1,∇ = d

dx).In the continuous case this discretisation is undone and we get back the derivatives, the kinetic


energy then becomes∫dnr a†(r)

(− ~2

2m∇2)a(r). In the interaction potential term we simply

replace discrete variables by continuous ones and sums by integrals. The overall result is

Hmult =

∫dnr a†(r)

(− ~2

2m∇2 + U(r)

)a(r)

+1

2

∫dnrdnr′Uint(r, r

′)a†(r)a†(r′)a(r′)a(r).

Historically this formula is the reason for the term ’second quantisation’. If we replace theoperator a(r) by the wavefunction ψ(r) and a†(r) by ψ∗(r) the noninteracting part of theHamiltonian looks like the expectation value of the energy in the state ψ(r),∫

dnr ψ∗(r)

(− ~2

2m∇2 + U(r)

)ψ(r).

Hence the noninteracting part looks as we had taken the wavefunction and turned it into aquantum operator, similarly to what is done with positions and momenta in ’first’ quantisation.For this reason a†(r), a(r) are often written as ψ†(r), ψ(r).

4.6 Fermions

So far we have defined creation and annihilation operators only for bosons. For fermions wehave to take into account that due to Pauli’s exclusion principle the only possible occupationnumbers are 0 and 1. However if the creation operator a†i is applied twice we would expect itto create two particles in state i and hence lead to a many-particle state violating the exclusionprinciple. The only way out is to demand (a†i )

2 = 0. If we introduce the anticommutator ofany two operators

[A,B]+ = AB +BA

this implies[a†i , a

†i ]+ = 0

reminding of the relation [a†i , a†i ] = 0 for bosons. Hence a mathematically satisfying defini-

tion of creation and annihilation operators for fermions might be one that yields all bosoncommutation relations (4.7) with commutators replaced by anticommutators:

[ai, aj ]+ = 0, [a†i , a†j ]+ = 0, [ai, a

†j ]+ = δij . (4.13)

A definition satisfying these conditions is the following

a†i | . . . ni = 0 . . .〉 = (−1)∑i−1k=1 nk | . . . ni = 1 . . .〉 (4.14)

a†i | . . . ni = 1 . . .〉 = 0 (4.15)

ai| . . . ni = 0 . . .〉 = 0 (4.16)

ai| . . . ni = 1 . . .〉 = (−1)∑i−1k=1 nk | . . . ni = 0 . . .〉 (4.17)

4.6. FERMIONS 43

where we have to consider only the cases that each operator acts on a state with ni = 0 orni = 1.

Motivation

• As expected (4.14) implies that a†i creates a particle in state i if it is applied to a many-particle state with ni = 0. Eq (4.17) indicates that the annihilation operator annihilatesa particle in state i if there is one to annihilate. The only aspect of these cases still tobe understood are the sign factors.

• Eqs. (4.15) and (4.16) are required in order to avoid particle numbers different from 0and 1. When applied to a state with a particle already present the creation operator maynot create a second particle so we obtain 0 instead. The annihilation operator appliedto a state without a corresponding particle present returns zero as states with ni = −1are not permitted.

• The signs in (4.14) and (4.17) are needed in order to satisfy the anticommutation rela-tions. To understand this imagine we wouldn’t have these signs. Then we would obtain,in the example of a system with two states,

a†1a†2|0, 0〉 = a†1|0, 1〉 = |1, 1〉

a†2a†1|0, 0〉 = a†2|1, 0〉 = |1, 1〉.

This contradicts the requirement a†1a†2 = −a†2a

†1. However in our example the requirement

is satisfied with the sign factors as in this case we have

a†1a†2|0, 0〉 = a†1|0, 1〉 = |1, 1〉

a†2a†1|0, 0〉 = a†2|1, 0〉 = −|1, 1〉.

• Let us now consider general a†i and a†j with i < j, and check that the above definition

satifies the correct anticommutation relation. To do so it is sufficient to apply a†ia†j and

a†ja†i to a general basis state with ni = nj = 0. Other basis states need not be considered

as we know that application of either product will map them to zero. For the basis statesunder consideration we obtain

a†ia†j | . . . ni = 0 . . . nj = 0 . . .〉

= a†i (−1)∑j−1k=1 nk | . . . ni = 0 . . . nj = 1 . . .〉

= (−1)∑i−1k=1 nk(−1)

∑j−1k=1 nk | . . . ni = 1 . . . nj = 1 . . .〉

and similarly

a†ja†i | . . . ni = 0 . . . nj = 0 . . .〉

= a†j(−1)∑i−1k=1 nk | . . . ni = 1 . . . nj = 0 . . .〉

= (−1)∑j−1k=1 nk+1(−1)

∑i−1k=1 nk | . . . ni = 1 . . . nj = 1 . . .〉.

In the second calculation the important difference is that a†j acts on a state where ni is1 rather than 0. As the summands nk in the formula are defined according to the initialbasis state the operators were applied to (with ni = 0) we thus have to add 1 to the sum

arising from a†j . As a result the states obtained by applying a†ia†j and a†ja

†i differ by a

sign factor, as desired.


• A full check of the commutation relations as well as the adjointness of ai and a†i is madein an exercise.

Chapter 5

Path integrals in secondquantisation

5.1 Bosons

We now want to discuss how to write down path integrals for many-particle systems in secondquantisation. For technical reasons we will not consider the full propagator, i.e. arbitrary

matrix elements of the time evolution operator e−i~ Ht , but only its trace. We will start by

discussing this trace in standard single-particle quantum mechanics, where it can be obtainedby integrating as in

tr e−i~ Ht =

∫dnr

⟨r∣∣e− i

~ Ht∣∣r⟩ =

∫dnrK(r, r, t).

If we were using a discrete basis we would instead compute the trace by summing over diagonalelements labelled by integers; here we are using a continuous basis and hence the sum has tobe replaced by an integral.

Motivation. To illustrate that the trace of the time-evolution operator is a natural quantityto look at we give two examples:

• We have seen that in Statistical Mechanics the partition function (for the canonicalensemble)

tr e−βH

plays an important role, and that the corresponding matrix elements can be accessedfrom the propagator through the replacement

i

~t→ β

For consistency with the treatment of the propagator we then continued to study thematrix elements but instead we could have looked at the trace.

• In Quantum Chaos the level density d(E) =∑

j δ(E−Ej) is obtained from the propagatorthrough a series of steps that involves taking a trace.

45

46 CHAPTER 5. PATH INTEGRALS IN SECOND QUANTISATION

Phase-space path integral for the trace. We have previously derived the Hamiltonianmechanics version of the path integral⟨

rf∣∣e− i

~ Ht∣∣r0

⟩=

∫D[r]D[p] exp

[i

~

∫ t

0dt′(p(t′) · r(t′)−H(r(t′),p(t′)

)]where r(0) = r0, r(t) = rf . Here the integration measure was defined by∫

D[r]D[p] . . . = limN→∞

1

(2π~)nN

∫dnr1 . . . d

nrN−1dnp0 . . . d

npN−1 . . .

To take the trace we have to identify r0 and rf and integrate over this variable as well. Thisleads to

tr e−i~ Ht =

∫dnr0

⟨r0

∣∣e− i~ Ht∣∣r0

⟩and thus

tr e−i~ Ht =

∫D[r]D[p] exp

[i

~

∫ t

0dt′(p(t′) · r(t′)−H(r(t′),p(t′)

)]. (5.1)

Here the integration measure has changed because we have absorbed the integral over r0 intoit. We now have∫

D[r]D[p] . . . = limN→∞

1

(2π~)nN

∫dnr0 . . . d

nrN−1dnp0 . . . d

npN−1 . . . .

As we have identified r0 and rf = rN the path integral is taken over all functions r(t′) withr(0) = r(t). One can show that it is permissible to also take the momenta as periodic withp(t) = p(0).1

As before the classical Hamiltonian H(r,p) used in the path integral is obtained from thequantum Hamiltonian by replacing the position and momentum operators r, p by the classicalvariables r, p.

Second quantisation. In second quantisation the Hamiltonian is written in terms of cre-ation and annihilation operators a†j and aj . It is hence natural to assume that a path integral

can be written down for such Hamiltonians as well, and that this will involve replacing a†j andaj by analogues of classical variables. This is indeed correct, and in the case of bosonic sys-tems the analogues of the classical variables are complex numbers that are mutually complexconjugate instead of adjoint. As we omitted writing hats above the creation and annihilationoperator the variable replacing aj will be just denoted by aj as well but it is now a complex

number. The creation operator a†j is replaced by the complex conjugate a∗j ∈ C.

We now expect an expression as in (5.1) where the Hamiltonian H inside the path integral hasthe creation and annilation operators replaced by the corresponding complex variables. Theonly thing that is not immediately clear is what will happen with the integral over p(t′) · r(t′).With the appropriate choice replacement for p(t′) · r(t′) the path integral turns into

1We can simply define pN = p0. Imposing such a definition would affect how time derivatives of p look in adiscretised formula but it is ok here as no such time derivatives enter the action.

5.1. BOSONS 47

tr e−i~ Ht =

∫D[a1, a2, . . .] exp

[ ∫ t

0dt′(−∑j

a∗j (t′)aj(t

′)

− i~H(a1(t′), a2(t′), . . . , a∗1(t′), a∗2(t′), . . .)

)](5.2)

Here we integrate over all complex functions aj(t′) subject to the condition aj(t) = aj(0).

These aj(t′) take the role of r(t′), p(t′) in (5.1).

The rest of this section will be shortened in the lecture. It is not examinable.

Motivation for the first term in the exponent. We have already given a heuristicexplanation for the way the Hamiltonian appears in Eq. (5.2). What happened to the termp(t′) · r(t′) looks less mysterious if we remember that the creation and annihilation operatorsare very similar to the ladder operators used when studying the harmonic oscillator in 3rd yearquantum mechanics. These operators were related to position and momentum operators. Ifwe assume that the same relation holds between aj(t

′), a∗j (t′) and fictitious coordinates rj(t

′)and momenta pj(t

′) we have

aj(t′) =

1√2~

(rj(t′) + ipj(t

′))

a∗j (t′) =

1√2~

(rj(t′)− ipj(t′))

(Compared to the formula in 3rd year quantum mechanics we have here set the m and ω equalto 1.) If we solve these equations for rj(t

′) and pj(t′) we obtain

rj(t′) =

√~2

(aj(t′) + a∗j (t

′))

pj(t′) =

√~2

1

i(aj(t

′)− a∗j (t′)).

Integration by parts then leads to the replacement

i

~

∫ t

0dt′pj(t

′)rj(t′) =

∫ t

0dt′

1

2(aj(t

′)− a∗j (t′))(aj(t′) + a∗j (t′)) = −

∫ t

0dt′a∗j (t

′)aj(t′) (5.3)

for each summand j.

Derivation. A full derivation of (5.2) is given e.g. in the Altland & Simons book. It makesuse of coherent states which we are not covering here.

Continuum limit. Similarly to section 2.4, if one takes a continuum limit of path integralsas here one obtains path integrals over a(x, t′) with x replacing the discrete index j. Thisalso applies if the system is naturally a continuous one and taking the continuum limit just


means avoiding a discrete approximation. For example for a one dimensional system withoutinteraction we would get

tr e−i~ Ht =

∫a(x,t′) with a(x,0)=a(x,t)

D[a] exp

[−∫ t

0dt′∫dx a∗(x, t′)a(x, t′)

− i~

∫ t

0dt′∫dx a∗(x, t′)

(− ~2

2m

d2

dx2+ U(x)

)a(x, t′)

]Here the first term in the exponent is what we get from (5.3) in the continuum limit. Thesecond term is what we get from the time integral over the Hamiltonian if we insert thenoninteracting part of Hmult given in the very end of section 4.5.

Operator ordering. In the derivation of the path integral in second quantisation one hasto pay attention at operator ordering. In general there are different ways to write the sameoperator, e.g. a†ia

†jajai and a†iai(a

†jaj − δij), and these will lead to different expressions if

we replace all operators by complex numbers. Indeed there are different conventions thatone can follow when deriving the path integral. The version in the Altland & Simons bookassumes normal ordering, i.e. all a†’s have to be placed before the a’s. Then one obtains theintegral as stated above, and when the action is written in discretised form the integral overthe Hamiltonian turns into a sum whose terms combine the a from each discretisation stepwith the a∗ from the following step. This difference is not visible in the continuous form of thepath integral.

Applications. The applications of path integrals in second quantisation are similar to thosefor single particle systems.

• Path integral are a convenient starting point for perturbation theory. For examplethe path integral for the Bose-Hubbard model with two sites can be written as

∫D[a1, a2, . . .] exp

−∫ t

0dt′∑j

a∗j (t′)aj(t

′)

− i

~

−a∗1(t′)a2(t′)− a∗1(t′)a2(t′) +U int

2

∑j

aj(t′)∗

2aj(t

′)2

.

If U int is small we can employ perturbation theory with a quadratic term composed ofthe leading summands and the interaction as a quartic perturbation.

• We can also look for contributions for which the action is stationary, as these playa role analogous to classical trajectories in single body quantum mechanics. For theBose-Hubbard model we can write the exponential as

ei~∫dt′f

where the integrand

f = −~i

∑j

a∗j aj + a∗1a2 + a∗1a2 −U int

2

∑j

a∗j2a2j

5.2. FERMIONS (NOT EXAMINABLE) 49

plays a role analogous to the Lagrangian. The action is stationary if the correspondingEuler-Lagrange equations are satisfied. Let us assume that these can be written downas if aj , a

∗j were independent variables (this requires a proof). Then we obtain

∂f

∂aj=

d

dt′∂f

∂aj,

∂f

∂a∗j=

d

dt′∂f

∂a∗j.

The second of these equation now boils down to

−~iaj + aj±1 − U inta∗j

2aj = 0

which can be rewritten as

i~aj = −aj±1 + U int|aj |2aj .

The first equation simply turns into the complex conjugate of this result. As the aj arosefrom replacing the wavefunction by an operator, and then replacing the operator againby a wavefunction, it is natural to compare their behaviour to that of a wavefunction.Without interaction the above equation would indeed turn into the Schrodinger equationfor the Bose-Hubbard model with a single particle. With the interaction we obtain asimilar equation where the interaction term gives a nonlinear contribution. It is calledthe nonlinear Schrodinger equation or Gross-Pitaevski equation. We will notconsider this equation further, but we note that its general variant plays an importantrole e.g. in the theory of Bose-Einstein condensation.

5.2 Fermions (not examinable)

This subsection is not examinable and will be shortened in the lecture.

We now want to generalise the preceding results and write down a path integral for fermionicsecond quantised systems. To do so we should again replace the creation and annihilationoperators by numbers. However this time we face the challenge that the annihilation andcreation operators anticommute (apart from the case [ai, a

†i ]+ = 1). In the bosonic case these

operators commuted (with an analogous exception) and it was therefore natural to replace themby complex numbers which also commute. In contrast for fermions we should use ’numbers’that anticommute. Such numbers have indeed been defined, and there are known as Grassmannvariables or simply Grassmannians.

Definition. η1, η2, . . . ηN are called Grassmann variables if they satisfy the commutationrelation

[ηi, ηj ]+ = 0

for all i, j = 1, 2, . . . , N or equivalently

ηiηj = −ηjηi.

Here we simply introduce variables defined to have these properties, we don’t require anyparticular representation e.g. through matrices. When applying these variables in the contextof creation and annihilation operators we also have to talk about complex conjugation ofGrassmannians, this issue will be postponed until later.


Corollary. As a consequence of our definition we have

ηiηi + ηiηi = 0

i.e.

η2i = 0

for all i.

Functions. As Grassmannians square to 0, all higher powers have to vanish as well. Hence ifwe consider general function of a single Grassmannian ηi its Taylor expansion will only involvethe constant and linear terms,

f(ηi) = a+ bηi.

For two Grassmannians we have the constant term, linear terms in both of them, and aquadratic term involving their product,

g(ηi, ηj) = a+ bηi + cηj + dηiηj .

For N variables we have 2N terms, each of which is either independent of or linear in each ofthe N Grassmannians.

In the language of Pure Mathematics, all possible functions of ηi, ηj (with coefficients in eitherR or C) form an associative algebra and the Grassmannians are the generators of this algebra.There is also a close link to the properties of differential forms studied in Fields, Forms andFlows.

Derivatives. Derivatives of these functions are easy to define as we just have to deal with theconstant and linear cases, but we have to note that the differential operator now anticommutesas well.

Definition. The derivative ∂∂ηi

is defined such that

• ∂∂ηiηi = 1,

• ∂∂ηi

applied to a term independent of ηi gives 0,

• ∂∂ηi

anticommutes with Grassmannian variables and other derivatives.

To realise why the anticommutation relation is required for consistency consider the followingderivative (for i 6= j)

∂

∂ηiηjηi = − ∂

∂ηiηiηj = −ηj .

Here we did not move the differential operator but we exchanged the two Grassmannians inorder to be able to apply the operator directly to ηi. However if we want to move ∂

∂ηito the

right instead we will obtain the same result only if it anticommutes with ηj .

Integrals. The definition of the integral is surprising at first.


Definition. The integral∫dηi is defined to coincide with the corresponding derivative, i.e.

•∫dηiηi = 1,

•∫dηi applied to a term independent of ηi gives 0,

•∫dηi anticommutes with Grassmannian variables and other derivatives/integrals.

We note that this integral does not have limits and should just be identified with a definiteintegral such as an integral from −∞ to ∞. There are no integrals with different integrationlimits in the Grassmannian context, and we should not expect differentiation and integrationto be inverses of each other. To motivate why this definition is natural we note that

• the result of the integral is independent of the integration variable, as expected for adefinite integral,

• and the result of Gaussian integrals is closely related to its conventional analogue.

We’ll now move on to compute the Gaussian integrals but first we need some more preparations.

Products of Grassmannians. Products of two Grassmannians commute with other Grass-mannians. This follows from

(ηiηj)ηk = −ηiηkηj = ηk(ηiηj)

where two minus signs cancelled, and it carries though immediately to products with an evennumber of factors.

Complex conjugation. One often has to apply Grassmannians in a context where thecorresponing commuting variables are complex. In this case it is helpful to introduce complexconjugation of Grassmannians. This is done by defining ηi and η∗i , but simply regarding themas independent Grassmannian variables.

(The conventions for this vary in the literature, and some authors refuse to use complexnotation for Grassmannians as there is no genuine complex structure. Some authors define(η∗i )

∗ = −ηi for reasons that we will not discuss here.)

One-dimensional Gauss integral. Using ’complex’ Grassmannians the Gauss integral canbe evaluated as ∫

dη∗∫dηe−aη

∗η

=

∫dη∗

∫dη(

1− aη∗η +1

2(aη∗η)2 + . . .︸︷︷︸

=0

)

=

∫dη∗

∫dη 1︸︷︷︸

=0

+a

∫dη∗

∫dη η︸︷︷︸=1

η∗

= a.


Here we Taylor expanded, used that powers of Grassmannians are zero and the integral of 1 iszero, then used the the anticommutation relation of η and η∗, and finally the definition of theintegral for linear terms. To compare this result to the case of commuting complex variableswe recall that (as seen in an exercise)∫

dRez dImz

π︸︷︷︸=dz∗dz

e−a|z|2

=1

π

√π

a

√π

a=

1

a

where we used twice the result for a real Gauss integral. With the formal definition of dz∗dzgiven here the results for the two cases are inverses of each other.

Multi-dimensional Gaussian integral. This observation carries over to the multi-dimensionalcase where we have, for commuting complex numbers∫

dz∗1dz1 . . . dz∗NdzNe

−z†Az =1

detA.

We will show that the corresponding Grassmannian integral is

I =

∫dη∗1dη1 . . . dη

∗NdηN exp

(−η†Aη

)= detA,

for arbitrary matrices A.

Proof: Writing the exponent in components we obtain

I =

∫dη∗1dη1 . . . dη

∗NdηN exp

−∑ij

η∗iAijηj

Here non-vanishing contributions can arise only from terms where each of the 2N Grassmannianvariables appears once in the integrand. These terms form part of the N -th term in the Taylorexpansion, which means that we have

I =

∫dη∗1dη1 . . . dη

∗NdηN

1

N !

−∑ij

η∗iAijηj

N

.

We can get rid of the minus sign if we exchange the η∗, η in the integration measure in a waythat produces a compensating factor (−1)N . This leads to

I =

∫dη1dη

∗1 . . . dηNdη

∗N

1

N !

∑ij

η∗iAijηj

N

Still a non-vanishing result will only occur for terms where each η∗i appears only once. Forthese terms it is natural to take the N factors and then reorder them with increasing i. Asthe factors each involve two Grassmannians they commute. But we have to take into accountthat each term with i written in increasing order arises from N ! terms in the line above. Thiscompensates the divisor N ! and we obtain

I =

∫dη1dη

∗1 . . . dηNdη

∗N

∑π

N∏i=1

η∗iAiπ(i)ηπ(i)


where we brought in a summation over permutations π that still has to be explained. To obtaina non-vanishing contribution also all choices of ηj have to be different. As both the i and jmust exhaust all integers from 1 to N they must be mapped to each other by a permutationπ. Hence we can replace j by π(i) if we take a sum over permutations π.

Finally we have to reorder the factors ηπ(i) such that they appear in increasing order as well.This can be done by exchanging η∗ factors between subsequent products η∗iAiπ(i)ηπ(i). Eachsuch exchange can be seen as a transposition, and it brings in a minus sign, as seen in thefollowing example:

η∗1η2η∗2η1 = η∗1η1η2η

∗2 = −η∗1η1η

∗2η2

Hence reordering the factors η leads to multiplication with the sign of π and we obtain

I =

∫dη1dη

∗1 . . . dηNdη

∗N

∑π

sgnπN∏i=1

η∗iAiπ(i)ηi.

Performing the integrals as in ∫dηidη

∗i η∗i ηi = 1

this leads to

I =∑π

N∏i=1

Aiπ(i) = detA

as desired.

Perturbation theory. We can now perform Gaussian averages in a way similar to commut-ing variables. A Gaussian average for Grassmannian variables should be defined by

〈. . .〉 =1

c

∫dη∗1dη1 . . . dη

∗Ndηne

−η†Aη . . .

where c = detA. We’ll just explicitly calculate the quadratic averages for the one dimension(A = a), to practice Grassmannian integration. We have

〈ηη∗〉 =1

a

∫dη∗dη η η∗e−aη

∗η =1

a

∫dη∗dη η η∗ =

1

a

where we only needed the constant term in the exponential as all later terms combine with theprefactor to quadratic or higher powers of the integration variables. In addition we have

〈ηη〉 = 〈η∗η∗〉 = 0

which follows immediately from the fact that the integrands are zero.

These examples motivate the following variant of Wick’s theorem (which we will not prove):

Wick’s theorem for Grassmannian variables. Averaged products of Grassmanniansηi, η

∗i can be evaluated as follows:

• Sum over all ways to contract the η’s and η∗’s pairwise.

• Reorder the factors such that contracted elements appear next to each other in the

form ηjη∗k (this may lead to sign factors from the anticommutation relation).

• Assign to each contraction line ηjη∗k a factor (A−1)jk.


As an example we have

〈ηj1η∗j2η∗j3ηj4〉 = 〈ηj1η∗j2η

∗j3ηj4〉+ 〈ηj1η∗j2η

∗j3ηj4〉 = −〈ηj1η∗j2ηj4η

∗j3〉+ 〈ηj1η∗j3ηj4η

∗j2〉

= −(A−1)j1j2(A−1)j4j3 + (A−1)j1j3(A−1)j4j2 .

Path integral. We are now ready to formulate a path integral for fermionic systems. Inanalogy to the bosonic case we have

tr e−i~ Ht =

∫D[a1, a2, . . .] exp

[ ∫ t

0dt′(−∑j

a∗j (t′)aj(t

′)

− i~H(a1(t′), a2(t′), . . . , a∗1(t′), a∗2(t′), . . .)

)]. (5.4)

Here the fermionic creation and annihilation operators have been replaced by (time-dependent)Grassmannians a∗j , aj . These are now subject to the condition

aj(t) = −aj(0), a∗(t) = −a∗(0).

In the derivation with coherent states (see Altland & Simons) this condition arises from theway the trace is written in terms of these states. The integration measure is given by

∫D[a1, a2, . . .] . . . = lim

N→∞

∏j

N−1∏k=0

∫da∗jkdajk . . .

where the index j runs over single particle states e.g. associated to sites and the index k =0, . . . , N − 1 runs over discretisation steps in time. All a∗jk, ajk are independent Grassmannianvariables.

Example. As an example we want to treat the fermionic harmonic oscillator. This isjust to illustrate the method, as we will see in the end for this simple system the result can beobtained much faster through direct computation. The Hamiltonian of the fermionic harmonicoscillator can be written as

H = ~ωa†a

where a†, a are fermionic creation and annihilation operators. Here we did not include a groundstate energy. The path integral now turns into

I = tr e−i~ Ht =

∫D[a] exp

[ ∫ t

0dt′(− a∗(t′)a(t′)− i

~~ωa∗(t′)a(t′)

)].

We will use the discretised form of this integral

I = limN→∞

∫da∗0da0 . . . exp

(−N−1∑k=0

a∗k+1

ak+1 − akτ

τ − iωN−1∑k=0

a∗k+1akτ

).

Here τ = tN is the width of the timesteps, and we wrote the Hamiltonian with the a∗ at

a later point in time than the a, in line with the earlier remark about operator ordering.


The antiperiodic boundary conditions in time entail aN = −a0 and a∗N = −a∗0. The aboveexpression can be rearranged as

I = limN→∞

∫da∗0da0 . . . exp

(−N−1∑k=0

a∗k+1ak+1 + (1− iωτ)N−1∑k=0

a∗k+1ak

)

= limN→∞

∫da∗0da0 . . . exp

(N−1∑k=0

aka∗k + (1− iωτ)

N−1∑k=0

a∗k+1ak

).

If we write a separate exponential for each summand and use that their Taylor expansion stopsafter the linear term we obtain

I = limN→∞

∫da∗0da0 . . .

N−1∏k=0

(1 + aka∗k)

N−1∏k=0

(1 + (1− iωτ)a∗k+1ak).

Now we will get a non-vanishing contribution only from terms where each ak and a∗k appearsas a factor exactly once. This is only possible if we either combine all linear terms from thefirst product or all linear terms from the second product, otherwise we would always miss aGrassmannian or have a Grassmannian appear twice. Hence I can be written as

I = I1 + I2

where

I1 = limN→∞

∫da∗0da0 . . .

N−1∏k=0

aka∗k = 1

and

I2 = limN→∞

∫da∗0da0da

∗1da1 . . . da

∗N−1daN−1

N−1∏k=0

(1− iωτ)a∗k+1ak.

To evaluate this integral it is helpful to permute da∗0 through to the end, and compensate thearising sign factor by writing da∗N . This leads to

I2 = limN→∞

∫(da0da

∗1)da1 . . . da

∗N−1(daN−1da

∗N )

N−1∏k=0

(1− iωτ)a∗k+1ak

where the arrangement of the integration variables mirrors that of the factors. This allows usto evaluate the integral as

I2 = limN→∞

(1− iωτ)N = limN→∞

(1− iωt

N

)N= e−iωt

leading to

I = tr e−i~ Ht = 1 + e−iωt.

This result can be checked immediately by direct calculation. The eigenstates of the fermionicharmonic oscillator are |0〉 and |1〉 with

~ωa†a|0〉 = 0, ~ωa†a|1〉 = ~ω|1〉.

This leads to the trace

tr e−i~ Ht = e−

i~E1t + e−

i~E2t = 1 + e−iωt

in agreement with the path integral result.

Advanced Quantum Theory - Sebastian...

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