A comment on functional integrals and many-body systems

Physica 97A (1979) 173-180 @ North-Holland

A COMMENT

Publishing Co.

ON FUNCTIONAL MANY-BODY

INTEGRALS

AND

SYSTEMS

Dalcio K. DACOL Department

of Physics,

University of California, Berkeley, CA 94720, ~$4

Received 28

November 1978

Green’s functions for systems of non-interactingparticles at T = 0 are obtained through ab initio calculations using functional integral algorithms. It is shown in detail how the correct Green’s functions are computed and how one recovers the structure of the ground state (which in the usual derivations is the starting point).

1. Introduction

Functional integration methods have an established reputation as a powerful and versatile tool in statistical mechanics’) and quantum field theory*). In this note we consider the use of such techniques in the study of the ground state of a many body system. We restrict ourselves to the case of noninteracting assemblies of bosons or fermions. Our purpose is to show in detail how one gets the correct Green’s functions and determines the structure of the ground state by carrying out an ab initio calculation using the functional integral algorithm for the Green’s function generating functiona13). The point here is to expose some ambiguities one is confronted with when taking as starting point the functional integral expression for the Green’s function generating functional and to show how one can get around such ambiguities. Below we study first a non-interacting system of spinless bosons and then a non-interacting system of spin S fermions. In both cases we use units such that h = 2m = 1, m is the particle mass.

2. The boson case The generating functional for the Green’s functions interacting bosons is given by3) Z[J] =

I I

of a system of non-

I% I% exp(iUcp, Jl) 3 Dv W expW[cp,

(1)

01) 173

174

D.K. DACOL

Ucp, Jl = here H(x)

I d4x[(P(x)(H(x)+ i~)(p(x)+J(x)(p(x)+(p(x)J(x)l,

= i(a/at)

convergence

factor

make the integrals We have

+ V’, (Y is an infinitesimal (exp[-(Y

J d4xQ(x)q(x)])

(2)

((Y+ 0’) which appears has been

introduced

because

a

in order

to

well defined.

to fix the boundary

conditions

satisfied

by the functions

p(x),

x = (x, t). Regalding the spatial variables, periodical boundary conditions are the natural choice for an extended system. For the time variable we assume p(x) = A = v(exp(i0)) = constant when t + +m (v z 0 and 0 s 8 < 27r). This is the most general

boundary

condition

we can choose

that allows

us to perform

partial integrations on the time variable without generating extra terms and preserving translation invariance. Now we change variables in the functional integrals by writing p(x) = A + a(x), with (T(X) = 0 when t -+ 2%. The constant A will be determined in terms of N, number of particles in the system, and V, the volume, by using standard prescriptions that relate the Green’s functions to observables4). In the case of a non-interacting system all observables can be expressed in terms of the two point Green’s function which, as will be shown, depends only on u, the magnitude of A. In order to see this recall the definition of the two point Green’s function: S2Z

G(x, y) = i

SJ(y)GJ(x)

J=O’

Now consider

the expression

Q(W)=/

Da DC? exp(il[A zz

+ (T, J])

(1 Da DC? exp(il[a,

Jl))exp(i

W[A, J]),

(4)

where W[A, .I] = J d4x[hs(x) + J(x)A]. Using Z = Q[A, J]lQ[A, 01 in (3) we find that G does not depend on the phase of A, a consequence of gauge invariance at .I = 0. Therefore account

all possible

the phase values

0 cannot

of 0. Thus

be determined

and we must

take into

we must write

2n

I

DP D@ exp(il[cp,

.I]) =

I 0

de

[I

Da DCTexp(iL[A

+ C, J])

I

?n = [I Da DC? exp(il[a,

.I])](

1 d0 exp(i W[A, J])).

(5) In this way we get

ON FUNCTIONAL

ZlJl

=[{I

L

~~~~:~]

175

INTEGRALS

(1/2a(

[

de

exp(iW]h, 51)).

(6)

The first term in eq. (6) is evaluated in a standard way. First we find f(x) such that (SL/G~)(,+, = 0, and f(x)].,+, = 0. Then we make another change of variables in the integrals by writing a(x) = f(x) + n(x), where n(x) is a new integration variable that obeys the same boundary conditions as u(x) (f(x) will vanish for large x if we assume J(x) to vanish for large x). With this change of variables we have L[a, J] = J d4x(J(x)f(x)) + L[n, 01. Thus Z[J] = [ exp( i 1 d”x(J(x)f(x))](l/2r)(

7 de exp(iW[h, 51)).

(7)

0

For f(x) we have (H(x) + ia)f(x) + J(x) = 0, and so (8)

f(x) = -I d4y[g(x, y)J(y)l, where g(x, y) = ((2~))~)

I

d4k[(ko - k2 + i(y)-‘] exp(ik(x - y)), kx = k . x - tko,

k2 = k . k,

d4k = d3k dk,,.

Using (3) we get for the Green’s function G(x,

Y) = -iv*+&,

Y),

(9)

which is the usual expression4) since we can show that v2 is equal to the particle density. We determine v2 by using the following relation4) N = i V((27rm4) [ Iii+ (1 d4kG(k) exp(ipko))], where G(k) is the 4-dimensional relation we get N=

Vu2 or

Fourier

transform

v’=N/V.

(10)

From G(x, y) we can deduce the momentum nP = (i/27r) [ hm ’ B-o+

of G(x, y). From this

(1

dpoW)

expW@o))]

distribution =

~~(277)~Wp)

through =

N&,0,

(11)

from which we infer that in the ground state of a non-interacting boson gas all particles are in the zero momentum state (n, is an expectation value, but from Z[J] one can show that the fluctuations about nP are zero).

D.K. DACOL

176

Let us discuss functional shows

space

that

the meaning

of the integration

over

the integrals

the set of all functions that

from

which

F can be decomposed a

degenerated

but since

in (1) are to be computed.

into a direct

cp(.u) such that

mathematical

over 0 in eq. (5). Let F be the

point

the degenerated

of

lim,,,,

sum of spaces

the

ground

states

are physically

(F(0)

This

p(x) = c(exp(i8))).

view

Eq. (5)

F(0)

state

is

is

means

infinitely

indistinguishable

(they differ from each other only by a phase factor) we must average over all of them. The origin of this degeneracy is the gauge invariance of the first kind exhibited by the system. We interpret this decomposition of F as a manifestation of a result due to Araki and Woods’) which shows that a cyclic representation of the canonical commutation relations describing a nonrelativistic boson gas is a direct integral of irreducible representations. Essentially the space F is associated with a cyclic representation (fixed by the particle density) and the space F(0) is associated with a corresponding irreducible representation also labelled by the angle 8.

3. The fermion

case

The generating of non-interacting

Z[J]

=.

\

I

functional for the Green’s functions spin S fermions is given by’)

in the case of a system

Dq, W, expW[cp,51) Dq, DG, expW[a

(12)

01)

Ucp, Jl = c 1 d4x[(Ps(xVf(xh(x) +~v(x)cp,(x)+ (P,,(x)J,.(x)I. s In the above components)

expressions cp< and J, (s is an index associated are non-commutative objects, i.e., they belong

with the spin to an infinite-

dimensional Grassmann algebra and the functional integration is performed on this algebra6). Due to the peculiar properties of the integration on such algebras no convergence factor is necessary. Regarding the boundary conditions satisfied by q,(x) we take them to be periodical boundary conditions for the spatial variables and for the time boundary condition we write lim,+, cps(x) = 0 (if we had chosen a non-zero value it would have to be a non-commutative constant which would have no physical interpretation). For Z[J] we get (by a procedure analogous to the one used in the boson case) Z[J]

= exp(i

T 1 d4x&(x)&(x)),

(13)

ON FUNCTIONAL

177

INTEGRALS

where H(x)f,(x)

+.I,@) = 0,

fs(X)lJ=O= 03

or f& I= -

cr j-d4yks& Y)UY)I,

with g&, Y>= [(27r-41 J d4k[g,,(k)1 exp(ik(x - y)), 1k) k) ~sr~k)~Ssr[ko-k’+i~~ko-k’-i~ I 'O', b(ko,

b(ko,

’

CY

b(ko, k) is an arbitrary function. The Green’s function is given by G,,(x, y) =

i

a?’ = g&, WY)6J,(X) I J=O

Y).

Now we have to determine b(ko, k). In order to do this we first introduce chemical potential for the fermion assembly and write

WL PI =

I I

a

I& D& exp(iE[cp, il) (I3 I%, I% exp(iE[cp, 01)’

Elcp,jl = Ucp, il+ p

I [s

1.

d4x x cPS(x)cp,(x)

By changing variables from cpS(x) to Q(X) = q,(x) exp(-ikt) and observing that the integration measure in (15) is invariant under such change of variables, we can show that G,,(x, x’) = i

6*wli, PI Si,(x’)~L(x)

exp(ip((t’ - t)).

(16)

I i=0 I

But i 62wti,cLl

&iM)VAx)

=

[(27rm411 d4k[hS,(k)] exp(ik(x -x’)>,

I i=~

h”(k) = “’ [ k,‘_-k~~~ :),

B(ko, k) ’ k. _ k? + cL_ i,

where again B(ko, k) is an arbitrary G,,(x, x’) we get G,,(k) = Mko - /.L,k),

function.

(Mko, k) = h,,(k)),

1’ For the Fourier

transform

of (17)

D.K. DACOL

178

Comparing

eq. (17) with g,,(k)

chemical

potential

henceforth Using

p. In fact

we will write standard

with the Green’s particles,

function

the energy

to know

b(F, k) instead

prescriptions

b(ko, k) is also a function

we see that we need

b only

of the

for k. = k’ therefore

of b(ko, k).

connecting

expectation

we get the following

and the momentum

values

expressions

distribution

of observable.<

for the number

of the fermion

of

assembly

N = (2s + 1)((23rF3) V 1 d3kb (F, k ), E = (2s + 1)((2~)

‘)V

I

d’k(k’b(F,

k)),

npr = b(pL, P).

(19)

of fermions Since nPr is the number exclusion principle and eq. (19) imply Osh(h,

k)c

in spin

state

r with

momentum

1.

(20)

In order to get an equation for b we use the following thermodynamic at T = 0 (K is the thermodynamic potential, K = E - pN):

N=-[e]i,r Thus

we must

p the

or

[1*$-$]~,,~=0.

relation

(21)

have

(2s + 1)((2rrm3)V

j- d’k(p

- k’) -& b(p, k) = 0.

Eq. (22) must be satisfied for all possible of N and V) therefore we must have (p-k’)g=O,

or

values

;)h=ci3(F-k’). alJ

of p (i.e. all possible

(22) values

(23)

thus b(p, k) = c,H(p -k’)

+ c$(k’-

p) + ~3.

(24)

where c, - cz = c and the constants c, must be independent of CL. Since the momentum integrals in eq. (18) must be finite we have to set c? = 0 and ci = 0. with (20), We have then b(p, k) = cH(p - k’) and so 0 4 b(p, k) s c, comparing we see that c = 1, therefore b(p, k) = 0(p -k’). Our final expression

for G,,,(k) is

(25)

ON FUNCTIONAL

G,,(k) = 6,,

e(ll. -k*) e(k2 - ‘) ko-k*+iu+ko-k’-ia

INTEGRALS

179

1’

which is the usual expression for the non-interacting fermion Green’s function4). Note also that eqs. (19) and (25) tell us that in the ground state all single particle energy levels from 0 up to a maximum value p are fully occupied and the levels above F are empty which, of course, is what we should get.

4. Conclusions As we showed in sections 2 and 3 the Green’s function in both cases is a distinct inverse of the operator H(x). In the boson case the appropriate inverse is determined by a prescription for avoiding the singularity at k. = k’. Such prescription was provided by the convergence factor in the functional integral. In the fermion case we do not have any prescription for avoiding this singularity which forces us to introduce an arbitrary function. By invoking the exclusion principle and appealing to a thermodynamic relation we succeeded in constructing the correct Green’s function for the fermion system. In both cases we did not make any explicit statements about the nature of the ground state. Our inputs were physically sound restrictions on the functional space over which the integrals in (1) and (12) are to be performed. Once the Green’s function generating functional is computed one can determine not only the Green’s functions but also the structure of the ground state. We must observe here that this procedure is just the opposite of what is done in standard derivations4) where one first determines the ground state and then proceeds to compute the Green’s function. Thus we see that it is possible to circumvent the ambiguities that appear when one tries to use the functional integral algorithms for ground state expectation values as a starting point in the study of many-body systems. Obviously for a non-interacting system such an approach is more complicated than the usual operator approach but one can speculate that perhaps in an interacting system, where the structure of the ground state is not known, the functional integral approach might be simpler and more direct, allowing us to bypass perturbation theory.

Acknowledgements

The author is grateful to Prof. H.L. Morrison for’interesting him in the use of functional integrals in quantum-statistical mechanics and to Rick Crewswick for an insightful discussion. This work was performed with financial

180

D.K.

DACOL

support from CAPES (an agency of the Brazilian federal government) form of a graduate studies fellowship.

in the

References 1) F.W. Wiegel, Phys. Rep. 16C (1975) 57. 2) E.S. Abers and B.W. Lee, Phys. Rep. 9C (1973) I. 3) H. Kleinert, Lectures at the 1977 Erice Summer School on Low Temperature published in Fortschr. Phys. 4) A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems New York, 1971). 5) H. Araki and E.J. Woods, J. Math. Phys. 4 (1963) 637. 6) F.A. Berezin, The Method of Second Quantization (Academic Press, London.

Physics,

to be

(McGraw-Hill,

1966).