Canonical Quantization and Path Integral Quantization

Trung Van Phan∗

Physics Department, Massachusetts Institute of Technology,

77 Massachusetts Avenue, Cambridge MA02139

Abstract

Operator-wise approach and functional-wise approach, Canonical Quantization and Path Integral

Quantization seems to be very different at first, but they indeed can be shown to be equivalent

with any nonsigular Lagrangian. For simplicity, let’s play around with real bosonic fields Φ first,

then if it’s worth, play with complex bosonic Γ and fermionic fields Θ later. The notes will deal

with this interesting relation in Quantum Field Theory at first, then later on we’ll play around

with no no goals.

1

I. CANONICAL QUANTIZATION ⇒ PATH INTEGRAL QUANTIZATION

With a given Lagrangian density L(Φ, ∂µΦ) of f bosonic fields in flat d = D+1 (D spatial

dimensions), the actionfunctional is defined to be SΩ[Φ] =∫

ΩddxL (inside a spacetime

volume Ω). This Lorentz-covariant expression can be used to read-off canonical formalism by

introducing the unit timelike vector Nµ = (1, 0, 0, 0), then one gets the canonical momentum

fields Πn and the Halmintonian density H in a time t slice σt (a spacelike hypersurface). For

simplicity, deal with nonsingular Lagrangian density:

Πn = ∂∂tΦnL = Nµ∂∂µΦL , H[Φ,∇Φ,Π] =

f∑n=1

Πn∂tΦn − L ; ∂Q =∂

∂Q(1)

The classical equations of motion can be read-off ∂tΦn = ∆ΠnH, ∂tΠn = −∆ΦnH; ∆Q = δδQ

.

To do Canonical Quantization, nominate (→) the fields and their canonical momentum

fields to Hermittian operators – note that these field operator are spacelike defined, as

Φn(x) → Φn(x) and Πn(x) → Πn(x). To generate dynamics, these fundamental operators

of the theory should have nontrivial commutation relation:

[Φn(x), Φm(x′)] = [Πn(x), Πm(x′)] = 0 , (2)

[Φn(x), Πm(x′)] = i~δn,mδ(d)(x− x′) ; ~→ 1 (3)

As shown above, let’s stick with Heisenberg’s picture of quantum for the rest of the notes

when dealing with Canonical Quantization since it seems to have a more intuitive realization

after the nomination. In this picture, the general physical state |Ψ〉 is time-independence,

while the fundamental operators and their eigenstates are time-dependence. Read-of the

dynamics from the equation of motion (dQ = ddQ

) for an operator O(x) (O(t = 0,x) = O(x)):

idtO(t,x) = [O(t,x), H[(t)]]→ O(t,x) = ei∫ t0 dτH[(τ)]Oxe−i

∫ t0 dτH[(τ)] , (4)

H[(t)] =

∫σt

dDxH[Φ(t,x),∇Φ(t,x), Π(t,x)] (5)

Therefore, the field eigenstates with fixed eigenconfigurations are changing with time. The

same goes for the canonical momentum field eigenstates:

Φn(t,x) = ei∫ t0 dτH[(τ)]Φn(x)e−i

∫ t0 dτH[(τ)] ⇒ |Φx, t〉 = ei

∫ t0 dτH[(τ)] |Φx〉 , (6)

Πn(t,x) = ei∫ t0 dτH[(τ)]Πn(x)e−i

∫ t0 dτH[(τ)] ⇒ |Πx, t〉 = ei

∫ t0 dτH[(τ)] |Πx〉 (7)

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To convert Canonical logic into Path Integral language, one has to make things functional.

The wavefunctional for state |Ψ〉 is defined to be Ψt[Φx] = 〈Φx, t|Ψ〉 = 〈Φx| e−i∫ t0 dτH[(τ)] |Ψ〉

in field-based, while Ψt[Πx] = 〈Πx, t|Ψ〉 = 〈Πx| e−i∫ t0 dτH[(τ)] |Ψ〉 in momentum-based. This

is a very interesting physical object that will be dealt with (in depth) in the later part of

the notes. The functional form of operators:

Φn(t,x) |Φx, t〉 = Φn(t, x) |Φx, t〉 , Πn(t,x) |Φx, t〉 = −i∆Φn(t,x) |Φx, t〉 (8)

Πn(t,x) |Πx, t〉 = Πn(t,x) |Πx, t〉 , Φn(t,x) |Πx, t〉 = −i∆Πn(t,x) |Φx, t〉 (9)

These can be used to show that (O(t = 0,x) = O(x)):

〈Φx|Πx〉 = 〈Φx, t|Πx, t〉 =√

2πfei

∑n

∫dDxΠ(x)Φ(x)=

√2πfei

∫dDxΠ(x).Φ(x)

(10)

Now, let’s choose a preference for field-based and start with an inequal-time inner product:

〈Φ′x, t′|Φx, t〉 =

∫ f∏n=1

dΠn,x 〈Φ′x, t′|Πx, t〉 〈Πx, t|Φx, t〉 (11)

=

∫DΠx 〈Φ′x, t′|Πx, t〉 〈Πx, t|Φx, t〉 (12)

=

∫DΠxe

−i∫ t′t dτH(τ) 〈Φ′x, t|Πx, t〉 〈Πx, t|Φx, t〉 (13)

=

∫DΠx

(2π)fei

( ∫dDxΠ(x).

(Φ′(x)−Φ(x)

)−∫ t′t dτH(τ)

)(14)

Now divide the time interval between t and t′ into m → ∞ subinterval of equal length

dt = t′−tn

for [tj, tj+1] (j = 0, 1, 2, ..., n), then the above expression becomes:

= limn→∞

∫ Φnx=Φ′x

Φ0x=Φx

( n−1∏j=1

DΦjx

)( n−1∏j=0

〈Φj+1x , tj+1|Φj

x, tj〉)

(15)

= limn→∞

∫ Φnx=Φ′x

Φ0x=Φx

( n−1∏j=1

DΦjx

)( n−1∏j=0

DΠjx

)( n−1∏j=0

〈Φj+1x , tj+1|Πj

x, tj〉 〈Πjx, tj|Φj

x, tj〉)

(16)

=limn→∞

(2π)f

∫ Φnx=Φ′x

Φ0x=Φx

( n−1∏j=1

DΦjx

)( n−1∏j=0

DΠjx

)ei∑n−1j=0

( ∫dDxΠj(x).

(Φj+1(x)−Φj(x)

)−H(τ)dτ

)(17)

=

∫ Φnx=Φ′x

Φ0x=Φx

D[Φx(τ)]D[Πx(τ)]ei∫ t′t dτ

∫dDx

(Πx(τ).∂τΦx(τ)−H[Φ(τ,x),∇Φ(τ,x),Π(τ,x)]

)(18)

=

∫ Φ′x,t′

Φx,t

D[Φ]D[Π]ei∫Ω d

dx

(Π(x).∂τΦ(x)−H[Φ(x),∇Φ(x),Π(x)]

); ∂Ω = σt + σt′ (19)

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Note that the subtlety with time-ordering is already included inside the definition of the

measurent D[Φ]D[Π]. From here, one can find the rule of calculating matrix element of

a time-ordered product and define the generating functional for general Z[J |Φ′x, t′; ,Φx, t]

(bubbles included) and connected W [J |Φ′x, t′; Φx, t] diagrams:

Z[J |Φ′x, t′; Φx, t] = eiW [J |Φ′x,t′;Φx,t] =

∫ Φ′x,t′

Φx,t

D[Φ]D[Π]ei∫Ω d

dx(Π.∂τΦ−H+J.Φ) ; (20)

i 〈Φ′x, t′|T(∏

j

Φnj ,x(tj))|Φx, t〉 =

(∏j

∆Jnj (tj)

)Z[J |Φ′x, t′; Φx, t]

∣∣∣J=0

, (21)

i 〈Φ′x, t′|T(∏

j

Φnj ,x(tj))|Φx, t〉C =

(∏j

∆Jnj (tj)

)W [J |Φ′x, t′; Φx, t]

∣∣∣J=0

(22)

Now, we have a functional language for the theory, but one can clearly see that it’s not

really the familiar Path Integral language – indeed, what we have is the phase-based Path

Integral. One has to integrate out the momentum conjugate fields in order to arrive at the

fields’ path only, and this can only be done if the Halmintonian density is ”nice enough”

(which is fortunately usually the case in Physics theories of interests). Note that although in

the classical theory (before nominating the classical fields into operators) there’s a relation

between Φn and Πn (canonical momentum field), but in the functional form this relation

isn’t clear, and indeed the integration has to do with every functional value for Πn and Φn

independently. That’s not surprising, since the direct starting point of Canonical Quanti-

zation is the Halmintonian density operator can be written in term of H[Φ,∇Φ, Π], and at

first glance it seems like Φn and Πn has their own freedom with no need for the pairs to be

canonical pairs. However, with a ”nice enough” Halmintonian, one can integrate out the

momentum conjugate field to arrives at a path integral (for the field functional) and put a

constrain on the leftover Πn in the exponential part, and interesting canonical interpreta-

tion pops out. Let’s do math to understand about this in a better light by considering the

following functional integration:

I[Φ] =

∫D[Π]e

i

( ∫Ω d

dxΠ.∂τΦ−H[Φ,∇Φ,Π]

)(23)

This is the weight for each Φ path. The ”nice enough” Halmintonian (usually the case of

interests in Physics) has the form (only terms with quadratic, linear or independence of Πn):

H[Φ(x),∇Φ(x),Π(x)] =1

2

∫dDxdDyΠn(t,x)Cnm[x,y; Φ(t,x)]Πm(t,y) +O(Π1,Π0) (24)

4

For local theory, Cnm[x,y; Φ(t,x)] = δ(x − y)Cnm[x; Φ(t,x)], then I[Φ] now becomes a

quasi-Gaussian integration that can be easily read-off:

I[Φ] =∣∣∣Cnm[x,y; Φ(t,x)]

∣∣∣− 12

Ωei

( ∫Ω d

dxΠ.∂τΦ−H[Φ,∇Φ,Π]

)S (25)

The stationary constrain impose the relation between Πn and Φn: ∂τΦn(τ,x) = ∆Πn(τ,x)H,

as Πn therefore has the canonical relation with Φn. Therefore, the phase can be intepreted

as the Lagrangian (and indeed, at classical limit, the path with minimum phase dominate,

which equivalent to the least action principle), hence only Φn and derivatives dependence:

I[Φ] =∣∣∣Cnm[x,y; Φ(t,x)]

∣∣∣− 12

Ωei

∫Ω d

dxL[Φ,∂µΦ] =(eδ

(d)(0)∫Ω d

dxTrln C[x;Φ])− 1

2ei

∫Ω d

dxL[Φ,∂µΦ] (26)

Therefore, I[Φ] = A[Φ]eiSΩ[Φ]. For C[x,Φ(x)] = C[x] independence on Φ, A[Φ] = A is just

a constant and it can be absorbed in the definition of D[Φ], which means the contribution

of different Φ path has the same norm (only the phase is different). At these special cases,

Canonical Quantization is equivalent to Path Integral Quantization. In general, the norm

of the path’s weight is path-dependence.

I’d like to thank Prof. L. Levitov for the discussion about the physical meaning of A[Φ].

II. PATH INTEGRAL QUANTIZATION ⇒ CANONICAL QUANTIZATION

Now, let’s go the other way around, from Path Integral Quantization back to Canonical

Quantization. Usually the calculation in Quantum Field Theory is done from Path Integral

side (since this approach is more intuitive than the other), and that makes aspects like

anomaly (quantum-classical mismatch) can be seen easier. By going backward, we will

learn about how these aspects can be realized in Canonical Quantization.

Path Integration Quantization is postulated from the inner product (transition ampli-

tude) between different fields’ configuration states as followed, in a functional form:

〈Φ′x, t′|Φx, t〉 =

∫ Φ′x,t′

Φx,t

D[Φ]eiSΩ[Φ] (27)

The Euler-Lagrangian can be used to read-off the classical behavior is straight-forward to

see, since the main contribution comes from the path with minimum phase, corresponding to

the classical least action principle. Let’s go back to Canonical Quantization by changing the

language to operator. Indeed, operator in Path Integration point of view is defined through

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the Path Integral matrix elements (non-0 only if t inside [t′, t′′], since outside of that range

there’s no clear definition of Φn,x(t) as the functional integration is ill-defined):

〈Φ′′x, t′′| Φn(t,x) |Φ′x, t′〉 =

∫ Φ′′x,t′′

Φ′x,t′D[Φ]Φn(t,x)eiSΩ[Φ] (28)

For any other functional to operator, the same trick applies. At each end, say t = t′, then

〈Φ′′x, t′′| Φn(t = t′,x) |Φ′x, t′〉 = Φ′n(t′,x) 〈Φ′′x, t′′|Φ′x, t′〉 for whatever on the other end, and

hence one gets Φn,x(t) |Φ′x, t〉 = Φ′n(t,x) |Φ′x, t〉, similar to the definition for operator acting

on state in Canonical Quantization.

Euler-Lagrangian equation of motion not only correct at the classical level, but also at

the quantum level. In fact, consider the matrix elements of the operator ∆Φn(t,x)SΩ[Φ]:

〈Φ′′x, t′′|∆Φn(t,x)SΩ[Φ] |Φ′x, t′〉 =

∫ Φ′′x,t′′

Φ′x,t′D[Φ]∆Φn(t,x)SΩ[Φ]eiSΩ[Φ] (29)

= −i∆Φn(t,x)

(∫ Φ′′x,t′′

Φ′x,t′D[Φ]eiSΩ[Φ]

)= −i∆Φn(t,x) 〈Φ′′x, t′′|Φ′x, t′〉 = 0 (30)

That’s true for t different from t′ and t′′ (and indeed Euler-Lagrangian comes from the

variation at hypersurfaces different from the 2 ends, since the configuration is kept fixed

there definitionally). For t = t′, one gets a totally different information from the boundary:

−i∆Φn(t,x) 〈Φ′′x, t′′|Φ′x, t〉 =

∫ Φ′′x,t′′

Φ′x,t

D[Φ]∆Φn(t,x)eiSΩ[Φ] (31)

=

∫ Φ′′x,t′′

Φ′x,t

D[Φ]δSΩ[Φ]

δΦn(t,x)]eiSΩ[Φ] (32)

=

∫ Φ′′x,t′′

Φ′x,t

D[Φ]

∫Ωddx∂µ

(∆∂µΦn(τ,x)LδΦn(τ,x)

)δΦn(t,x)

eiSΩ[Φ] (33)

=

∫ Φ′′x,t′′

Φ′x,t

D[Φ]

∫σtdDx∆∂τΦn(t,x)LδΦn(t,x)

δΦn(t,x)eiSΩ[Φ] (34)

=

∫ Φ′′x,t′′

Φ′x,t

D[Φ]∆∂τΦn(t,x)LeiSΩ[Φ] (35)

=

∫ Φ′′x,t′′

Φ′x,t

D[Φ]Πn(t,x)eiSΩ[Φ] = 〈Φ′′x, t′′| Πn(t,x) |Φ′x, t〉 (36)

Πn,x(t) is nothing but the canonical momentum field of the field Φn(t,x), and from Path

Integration one see that in field-based Πn(x) → −i∆Φn(x), therefore one arrives at the

commutation relation similar to Canonical Quantization (note that all operators in their

functional forms are right-all – act on all object on the right side). But that’s not the whole

6

story, isn’t it? Πn can be written down in terms of Φn and derivatives from the Lagrangian, so

it’s already has its functional field-based form. So, where really is the canonical commutation

relation? The truth is, you have to blame the non-equal time ∂t operator (thanks for the

subtlety of the time ordering), since it’s what mess things up. The functionals Φn and

∇Φn are well-defined in every hypersurface, but not ∂tΦn (must be included inside Πn) –

it’s indeed not a functional but an operator that deal with the communication between

hypersurface (and it’s highly non-trivial). Therefore, let’s try to reproduce by seperating

things inside a composite functional by an infinitesimal time (different way to order these

in time corresponds to the ordering problem), heurestically:∫ Φ′′x,t′′

Φ′x,t′D[Φ]Πn(t,x)Φn(t,x)eiSΩ[Φ] =

∫ Φ′′x,t′′

Φ′x,t′D[Φ]Πn(t,x)Φn(t− ε

2,x)eiSΩ[Φ] (37)

or

∫ Φ′′x,t′′

Φ′x,t′D[Φ]Φn(t+

ε

2,x)Πn(t,x)eiSΩ[Φ] (38)

⇒∫ Φ′′x,t

′′

Φ′x,t′D[Φ][Πn(t,x),Φn(t,x)]eiSΩ[Φ] = −ε

∫ Φ′′x,t′′

Φ′x,t′D[Φ]Πn(t,x)∂tΦn(t,x)eiSΩ[Φ] (39)

= −∫ Φ′′x,t

′′

Φ′x,t′D[Φ6=t]

(D[Πn] + ...

)(Π2n(t,x) + ...

)ei

∫Ω d

dx( 12

Π2n+...) = −i (40)

Up there, we use Πn = ∂tΦn + ... and D[Πn] = D[εΦn] + ... (neighbor slices), which is

expected from a diagonal Π-quadratic theory (more complicated theories can be generalized

similarly). Then the quasi-Gaussian functional integration can be done.

In short, at this point it’s clear that Path Integration Quantization can lead to Canonical

Quantization – different than the other way, since Canonical Quantization leads to Path

Integration Quantization only in some special cases. Canonical Quantization leads to phase-

based Path Integration Quantization, however.

III. WAVEFUNCTIONAL

To find the ground state wavefunctional, one needs to find the lowest energy state config-

uration, which means it’s necessary to find the lowest value in the spectrum of H[(t)] (say,

E0) and the corresponds eigenfunctional 〈Φx, t|Ψ0, t〉 (field-based):

〈Φx, t| H[Φ(t,x),∇Φ(t,x), Π(t,x)] |Ψ0, t〉 = E0 〈Φx|Ψ0〉 = E0Ψ0[Φx] (41)

= H[Φ(t,x),∇Φ(t,x),∆Π(t,x)] 〈Φx, t|Ψ0, t〉 = H[Φ(t,x),∇Φ(t,x),∆Π(t,x)]Ψ0[Φx] (42)

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To normalize the ground state, imposing:

1 = 〈Ψ0|Ψ0〉 =

∫DΦx 〈Ψ0|Φx〉 〈Φx|Ψ0〉 =

∫DΦx

∣∣∣Ψ0[Φx]∣∣∣2 (43)

The same treatment can be used for higher excited states. For vacuum-vacuum transition

amplitude (different time slices means different eigenstates):

〈Ψ0, t′|Ψ0, t〉 =

∫DΦ′xDΦx 〈Ψ0, t′|Φ′x, t′〉 〈Φ′x, t′|Φx, t〉 〈Φx, t|Ψ0, t〉 (44)

=

∫DΦ′xDΦxΨ0[Φ′x]Ψ0[Φx]∗

∫ Φ′x,t′

Φx,t

D[Φ′′]eiSΩ[Φ′′] (45)

By infinitesimal tilting the time axis by a factor (1− iε), any state will evolve to the ground

state (E0 < Ej) at the far past (t) or future (t′) up to a prefactor; |εE0t|, |εE0t′| 1:

|Φx, t〉 = limε→0|Φx, t+ iε〉 = lim

ε→0

( eitE0

e−εtE0〈Ψ0|Φx〉 |Ψ0〉+

∑j

eitEj

e−εtEj〈Ψj|Φx〉 |Ψj〉

)(46)

⇒ |Φx, t〉 ≈ limε→0

eitE0

e−εtE0〈Ψ0|Φx〉 |Ψ0〉 → |Ψ0, t〉 = lim

ε→0

e−εtE0

〈Ψ0|Φx〉|Φx, t〉 , (47)

〈Φ′x, t′| = limε→0〈Φ′x, t′ + iε| = lim

ε→0

(e−it′E0

eεt′E0〈Φ′x|Ψ0〉 〈Ψ0|+

∑j

e−itEj

eεtEj〈Φ′x|Ψj〉 〈Ψj|

)(48)

⇒ 〈Φ′x, t′| ≈ limε→0

e−it′E0

eεt′E0〈Φ′x|Ψ0〉 〈Ψ0| → |Ψ0, t′〉 = lim

ε→0

eεt′E0

〈Φ′x|Ψ0〉〈Φ′x, t′| (49)

Therefore, one can guess:

〈Ψ0, t′ → +∞|Ψ0, t→ −∞〉 = limε→0

eε(t′−t)E0

〈Ψ0|Φx〉 〈Φ′x|Ψ0〉〈Φ′x, t′|Φx, t〉

∣∣∣∣∣t′→+∞,t→−∞

(50)

= limε→0

eε(t′−t)E0

Ψ0[Φ′x]Ψ0[Φx]∗

∫ Φ′x,t′

Φx,t

D[Φ′′]eiSΩ[Φ′′]

∣∣∣∣∣t′→+∞,t→−∞

(51)

Strangely, the answer doesn’t seem to depend on the boundary field configuration. Hence,

the best guess for the vacuum-vacuum transition amplitude (obviously, I don’t buy it):

〈Ψ0,+∞|Ψ0,−∞〉 ∼∫D[Φ]eiSΩ[Φ] (52)

The most conventional trick to get the above relation is based on the assumption that

Halmintonian also time-dependence, and the theory becomes free at distance past and future

(although physically it seems to be nonsense). Compare to the tilting trick, this method

is more convincing. Mathematically, Ψ0,±∞[Φx] contribution can be calculated to be an

infinitesimal part in the fields’ quadratic pieces of the Lagrangian – the famous iε inside

m2 + iε (ε→ 0+). See [2], Chapter 9.2 for details.

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IV. CURVED SPACETIME QUANTIZATION

V. PARTICLE INTEPRETATION

Single particle intepretation, Fock’s space, field renormalization, general renormalization,

anomaly

VI. APPENDIX

A. Spacelike Hypersurface for Canonical Formalism

Indeed, only a spacelike hypersurface can provide initial conditions for time evolution. If

the initial hypersurface is not spacelike, then some events on this surface will be in causal

contact with each other, implying that the field values cannot be chosen independently.

This violates the requirement of phase space that the variables may be chosen freely on

an initial hypersurface. It’s not only a bad idea to do quantization on a non-spacelike

surface, it’s impossible to formulate a well-posed initial value problem. This is true of both

classical and quantum physics. There’s also a technical problem with a non-spacelike initial

hypersurface: the Laplace operator is no longer elliptic, meaning that it will have a spectrum

with singularities, and cannot be inverted.

Another way to put it is as follow: canonical quantization can only make sense on a

spacelike hypersurface. Typically, a slice of constant time is a spacelike hypersurface, but

sometimes it’s not (the case of Schwarzschild time is a good example). Strictly speaking,

there’s no need to use a slice of constant time, just a spacelike hypersurface which can be

defined as a space of constant time in appropriate coordinates. Quantizing on a timelike

hypersurface means trying to quantize on a surface of constant position in space there, which

has to be bad.

I’d like to thank Prof. E. Bertschinger and J. Bloomfield for this explanation.

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B. Problem with nonsingular Lagrangian

VII. REFERENCES

[1] M. MASUJIMA: Path Integral Quantization and Stochastic Quantization. Springer -

Verlag Berlin Heidelberg, 2000

[2] S. WEINBERG: The Quantum Theory of Fields, Vol 1: Foundation. Cambridge

University Press, 1996

∗ Electronic address: phanvltt@mit.edu

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