Rajdeep Sensarma - Theoretical Physics (TIFR) …theory.tifr.res.in › ... › courses ›...

Advanced Quantum Mechanics

Lecture #24

Path Integrals and QM

Rajdeep Sensarma [email protected]

Recap of Last Class

• Replica Trick and Linked Cluster Theorem

• Two point correlators in Gaussian Theory : Matsubara Frequencies

• Evaluating Matsubara sums: 1st order corrections to Z

• Rotation to real frequencies ——> poles of time-ordered functions

• Perturbation Expansion for 2-point function

Path Integrals for Many-Particle SystemsLet us recap how we obtained path integral formalism for single particle QM

• Start with matrix element of time evolution operator U = e-iHt between position eigenstates

• Break it up into products of matrix elements of infinitesimal evolution operator between intermediate position eigenkets. Obtain a large number of integrals over intermediate co-ord.

U(xf , tf ;xi, ti) =

Zdx1dx2..dxN�1 [

hxf |U(tf , ti+(N�1)✏)|xN�1ihxN�1|U(ti+(N�1)✏, ti+(N�2)✏)|xN�2ihxN�2|...

|x2ihx2|U(ti + 2✏, ti + ✏)|x1ihx1|U(ti + ✏, ti)|xii]

• For infinitesimal matrix elements, introduce momentum states [Note H=H(x,p)] and use normal ordering to convert this to exponential of the infinitesimal Lagrangian.

• Work out momentum integrals, take continuum limit, to get

S[x(t)] =

Z tf

ti

dt

1

2mx

2(t)� V (x(t))

�=

Z tf

ti

dtL[x(t), x(t)]

U(xf

, t

f

;xi

, t

i

) =

✓mN

2⇡i(tf

� t

i

)

◆1/2 ZD[x(t)]eiS[x(t)]

Path Integrals for Many-Particle SystemsFor many particle systems, the natural operators to write the Hamiltonian are creation/annihilation operators, i.e. H = H(a†, a). These are the conjugate operators with (anti) commutation relations.

We can simply follow the derivation for one particle QM, if we can find the eigenstates of the creation/annihilation operators and introduce resolution of identity with these states.

The many-body basis that we have seen till now is the occupation no. basis ——> not eigenstates of a†, a. Search for new basis ———> Many Body Coherent States

The equivalent of Path Integrals, written in these bases, are called functional integrals

Coherent StatesRecap of Harmonic Osc.:

Coherent states are defined as the right eigenstates of the annihilation operator

Corresponding to |ξ>,

•Linear Comb. of diff. number states:

•Orthogonality: •(Over)completeness:

•Linear Dependence:

The coherent states can still be used as basis set to obtain unique expansion of an arbitrary state, as long as we restrict the expansion co-efficients to be of the form

f is a fn. of only ξ* and not of ξ

Many-Body Coherent States Let us generalize the defn. of a coherent state as the right eigenstate of all the annihilation operators, one for each single-particle basis state.

∀ α

We will immediately see that coherent states for Bosons and Fermions have very different description.

Bosons: φα is a complex number

Fermions: φα and φβ anticommute

Grassmann Numbers

Will not work with Grassmann Numbers in this course

a↵|�i = �↵|�i

[a↵, a� ]|�i = 0 ) [�↵,�� ] = 0

{a↵, a�}|�i = 0 ) {�↵,��} = 0

Bosonic Coherent States A Bosonic coherent state is the right eigenstate of all the annihilation operators, one for each single-particle basis state.

∀ α where φα is a complex number

Similarly the left eigenstate of all the creation operators, one for each single-particle basis state.

Expansion in occupation no. basis: Clearly

∀ α

This is an un-normalized state. However with these states, one has the relations

a↵|�i = �↵|�i

a↵|�i = �↵|�i

h�|a†↵ = h�|�⇤↵

|�i =X

{n↵}

�n↵1↵1 �

n↵2↵2p

n↵1 !n↵2 !..|n↵1 , n↵1 ..i

=X

{n↵}

Y

↵

(�↵a†↵)n↵

n↵!|0i = e

P↵ �↵a†

↵ |0i

a†↵|�i =@

@�↵|�i h�|a↵ =

@

@�⇤↵

h�|

Bosonic Coherent States (Non) Orthogonality of Bosonic Coherent States:

Extension of SHO coherent states with different normalization.= e

P↵ �

0⇤↵ �↵

Note: In each term, each n𝛼 = nβ for the expectation to be non zero. For n𝛼 = nβ, one factor of

n𝛼! is cancelled by the matrix elements of (aa†)n𝛼

=X

{n↵}

Y

↵

(�0⇤↵ �↵)n↵

n↵!

= h0|X

{n↵}

Y

↵

(�0⇤↵ a↵)n↵

n↵!

X

{n�}

Y

�

(��a†�)

n�

n� !|0i

h�0|�i = h0|e

P↵ �

0⇤↵ a↵e

P� ��a

†� |0i

Bosonic Coherent States Resolution of identity: in Fock Space

where

Trace of an operator A

Tr A =X

n

hn|A|ni =

Z Y

↵

d�⇤↵d�↵

2⇡ie�

P↵ �⇤

↵�↵X

n

hn|�ih�|A|ni

=

Z Y

↵

d�⇤↵d�↵

2⇡ie�

P↵ �⇤

↵�↵h�|A X

n

|nihn|!|�i

=

Z Y

↵

d�⇤↵d�↵

2⇡ie�

P↵ �⇤

↵�↵h�|A|�i

For a state |φ> in Fock space

wavefunction

| i =Z Y

↵

d�⇤↵d�↵2⇡i

e�P

↵ �⇤↵�↵ |�ih�| i =

Z Y

↵

d�⇤↵d�↵2⇡i

e�P

↵ �⇤↵�↵ (�⇤)|�i

d�⇤↵d�↵

2⇡i=

d(Re�↵)d(Im�↵)

⇡

Z Y

↵

d�⇤↵d�↵

2⇡ie�

P↵ �⇤

↵�↵ |�ih�| = 1

Bosonic Coherent States Matrix Elements:

Schrodinger Equation: H(a†↵, a↵)|�i = E|�i ! H

✓�⇤↵,

@

@�⇤↵

◆ (�⇤) = E (�⇤)

Normal Ordered Operators: An operator where all creation operators occur to the left of all annihilation operators

: a†x

ax

a†y

ay

:= a†x

a†y

ax

ay

For normal ordered operators, h�|A(a†↵, a↵)|�0i = A(�⇤

↵,�0

↵)

a↵ ! @

@�⇤↵

a†↵ ! �⇤↵

h�|a†↵|fi = �⇤↵f(�

⇤)h�|a†↵|fi = �⇤↵f(�

⇤)

Functional Integrals for many Bosons

Start with the matrix element of the time evolution operator between two coherent states.

Since any state in Fock space can be expanded in coherent state basis.

U(�f , tf ;�i, ti) = h�f |e�iH(a†,a)t|�ii

= h�f |N�1Y

n=0

e�iH(a†,a)(tn+1�tn)|�ii t0 = ti, tN = tf, tn = ti+nε

Break up into products of large no. of infinitesimal evolution operators

Remember that for each time index, we have a product over single particle basis 𝛼

Introduce φ𝛼(t) where t takes discrete values on the lattice t0 = ti, tN = tf, tn = ti+nε

E.g. if 𝛼 is the position basis x, we can use φ(x,t)

Introduce resolution of identity with many-body coherent states.

between each infinitesimal operator

Z Y

↵

d�⇤↵d�↵

2⇡ie�

P↵ �⇤

↵�↵ |�ih�| = 1

The Infinitesimal Evolution operatorUnlike the case of single particle Hamiltonian H =

p2

2m+ V (x)

where H is already normal ordered, and Trotter errors in e-iHε arise in 2nd order in ε

the many body H is not necessarily normal ordered

H =X

x

a†x

�r2

2max

+X

xx

0

V (x� x0)a†x

ax

a†x

0ax

0E.g. :

So, it is important to start with a normal ordered form of the many-body H

H =X

x

a†x

�r2

2m+ V (x� x0 = 0)

�ax

+X

xx

0

V (x� x0)a†x

a†x

0ax

0ax

From now on, we will assume that many-particle Hamiltonians are written in their normal ordered forms.

With this assumption, the Trotter errors can be neglected as they are O(N✏2) ! 0

in the continuum limit

Functional Integral for BosonsTaking all these together, we have

h�n+1|e�iH(a†↵,a↵)✏|�ni = e�iH(�⇤

↵(n+1),�↵(n))✏h�n+1||�ni+O✏2

Remember that the resolution of identity at tn+1 gives an additional factore�

P↵ �⇤

↵(tn+1)�↵(tn+1)

Let us now collect them together to get

e�P

↵ �⇤↵(tn+1)[�↵(tn+1)��↵(tn)]�i✏H[�⇤

↵(tn+1),�↵(tn)]

= ei✏n

P

↵ �⇤↵(tn+1)

h

i�↵(tn+1)��↵(tn)

✏

i

�H[�⇤↵(tn+1),�↵(tn)]

o

Note that there are N matrix elements, but N-1 insertion of identity resolutions. So all the terms cannot be paired up in this way.

Choice: Keep the eP

↵ �⇤↵(tN )�↵(tN�1) coming from the final matrix element hanging around

= e�iH(�⇤↵(tn+1),�↵(tn))✏e

P↵ �⇤

↵(tn+1)�↵(tn) +O✏2

Note: φ* always appears at a time point εshifted from the time pt. at which φ appears

Functional Integral for Bosons

U(�f , tf ;�i, ti) =

N�1Y

m=1

Y

↵

Zd�⇤

↵(tm)�↵(tm)

2⇡i

!eP

↵ �⇤↵(tf )�↵(tN�1)

ei✏

PN�1m=1

n

P

↵ �⇤↵(tm+1)

h

i�↵(tm+1)��↵(tm)

✏

i

�H[�⇤↵(tm+1),�↵(tm)]

o

Continuum Limit :

dt i ∂tφα

Remember, by convention φ*(t) =φ*(t), and φ(t) =φ(t-ε)

N�1Y

m=1

Y

↵

Zd�⇤

↵(tm)�↵(tm)

2⇡i

!=

ZD[�⇤

↵(t),�↵(t)]Short hand notation:

U(�f , tf ;�i, ti) =

ZD[�⇤

↵(t),�↵(t)]eP

↵ �⇤↵(tf )�↵(tf )eiS

S =X

↵

Z tf

ti

dt {�⇤↵(t)i@t�↵(t)�H[�⇤

↵(t),�↵(t)]}where the action

Functional Integral for BosonsFor specificity, let us choose the single particle basis α to be the position basis

S =

Z tf

ti

dt

Zd3x {�⇤(x, t)(i@t)�(x, t)�H[�⇤(x, t),�(x, t)]}

U(�f

, t

f

;�i

, t

i

) =

ZD[�⇤(x, t),�(x, t)] e

Rd

3x�

⇤(x,tf )�(x,tf )e

iS

φ*(x,t) and φ(x,t) are “operator valued” functions of space and time. These are the quantum fields.

Lagrangian density

The action is now a space time integrated object (no notion of “paths”). The integrand is the Lagrangian density of the system.

The matrix element of the time evolution operator is a functional integral over all possible field configurations, with each configuration contributing eiS (times the boundary term), where S is the action written in terms of the fields and their derivatives wrt space and time.

In this way of approaching many-particle QM, the external input is the nature of the fields ( bosonic or fermionic, spin —> multiple fields, bosonic+fermionic etc.) and the action written in terms of these fields. The output is the propagator and related quantities.

Weakly Repulsive Spin 0 Bose GasThe Hamiltonian for a repulsive Bose gas interacting with local (delta fn) interaction potential.

(normal ordered form)

H =

Zd3xa†

x

✓�r2

2m� µ

◆ax

+ ga†x

a†x

ax

ax The chemical potential absorbs quadratic

terms coming from normal ordering

=

Zd3ka†k

✓k2

2m� µ

◆ak +

g

V

Zd3k

Zd3k0

Zd3q a†ka

†k0ak0�qak+q

Introduce Bosonic fields: φ*(x,t) and φ(x,t) or φ*(k,t) and φ(k,t)

The action for this system is given by

S =

Z tf

ti

dt

Zd

3x

⇢�

⇤(x, t)

✓i@t +

r2

2m+ µ

◆�(x, t) � g �

⇤(x, t)�⇤(x, t)�(x, t)�(x, t)

�

U(�f

, t

f

;�i

, t

i

) =

ZD[�⇤(x, t),�(x, t)] e

Rd

3x�

⇤(x,tf )�(x,tf )e

iS

A quadratic/gaussian theory is obtained in the non-interacting limit.

Rajdeep Sensarma - Theoretical Physics (TIFR) …theory.tifr.res.in › ... › courses ›...

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