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Spectral representation of the many-time, causal Green's function in nonrelativistic many-body theory
ANNALS
OF
Spectral
PHYSICS:
19,
448-457 (1962)
Representation of the Function in Nonrelativistic Don-sun School
of Physics,
University
H.
Many-Time, Many-Body
Causal Green’s Theory*?
E(OBEf
of Minnesota,
Minneapolis
14,
Minnesota
The exact spectral representation of the many-time, causal Green’s function, which is the ground state average of the time ordered product of creation and annihilation operators in the Heisenberg picture, is obtained in this paper by taking the time Fourier transform of the function. This spectral representation shows that the function has poles at the excitation energies, and residues which are related to the st,at,e amplitudes. The usual two-time Green’s function can be written in terms of the many-time Green’s function, and the twotime spectral representat,ion obtained from the many-t,ime spectral representation by a process of integration. The two-time spectral represent,ation obtained in this manner agrees with the one obtained directly, but the derivation gives some insight into the relation between the two spectral representations. It is shown that in special cases the general two-time spectral representation obtained here is essentially the same as those obtained by others. I. INTROI>UCTION
The modern field t#heoretir technique of Green’s functions has become very important, in the last, few years in t#hc many-body problem ( 1, 2). I+?-omthe Green’s functions, which are averages of the time ordered product of field operators in the Heisenberg picture, the energy spectrum, information about, the state amplitudes, and even t#hct,hcrmodynamic proptr%ies of t’he system can be obt,ained (5, 2, 1). The Green’s fun&ions describe the creat,ion, propagation with consequent mutual and self-interactions, and subsequent annihilation of the particles. The t’wo-time causal, retarded, and advanced Green’s functions have been the most widely used C,$), since t’heir spectral reprcsentat8ions are well known. The spectral represent#ation of the two-time Green’s function, which can be obtained exactly, shows explicitly t’hat the poles of the funct’ion are t,he excitation energiesand the residuesare st#atcamplitudes. * This paper is based on part of a t,hesis submitted in parGal fulfillment of the requirements for the degree of I>octor of Philosophy at the University of Minnesota. t Supported by the U. S. .Stomic Energy Commission. $ Present address: Department of Physics and iZstronomy, The Ohio State University, Columbus 10, Ohio.
448
SPECTRAL
REPRESENTATIOS
448
The many-time causal Green’s functions have not been used as much in the many-body problem, because among other reasons the general spectral representations are not known (4). The spectral representat,ions of three-time t,hermal Green’s funct)ions have been studied by Bench-Bruevich (5) who also makes some general remarks about the many-t,ime functions. It, is the purpose of t#his paper to obt)ain the exact spectral representat8ion of the many-time, causal Green’s funct8ion and show its relation t,o t,he spect,ral representation of the twotime Green’s function. The spectral reprcsent,ation of Green’s functions or propagat,ors in rclat,ivistic field theory were first, obtained by Lchmann (6) and are thus known as Lehmann expansions. Nambu (7) has studied the structure of relativistic Green’s functions and obtained spectral representations which are parametric integrals over invariant energy denominators. However, he derived the representations on t.he basis of perturbation theory. III the nonrelnt,ivist,ic many-body problem, there is no Lorcntz invariance, so the relativistic speckal rcprcscnt,ations do not apply. For this case, exact spectral rcprcscntat8ions were first obtained by Bench-Brucvich (8) for the one and two particle Green’s funct,ions. He and Kogan (3) later extended this work to include general t,wo-t,ime temperat,urr dependent Green’s functions. Landau (9) has obtained a spectral representation of t)he single part,iclt, temperature dependent Green’s function, and used it, t,o derive a dispersion relation between the real and imaginary parts of the function. Bolsterli (10) has used t,he one and two particlc Green’s functions for systems of fcrmions with pairing forces to obt,ain the pairing energy, t,he chemical potential, and the excitation energies from an examination of their poles. The speckal representation of the single particle Green’s function has also been studied by Galit,skii and Migdal (2). A good review of twotime t,cmpcrat,ure dependent Green’s functions has been given by Zubarev (4) and he shows why the spectral representations of the retarded and advanced Green’s functions ( 11) are so useful. In Scct,ion II of this paper the many-time causal Green’s function is defined in coordinate and time space. Its Icourier coeficicnt is the Green’s function in momentum and time space, which can be rewritten in a more convenient form for taking t,hc t,imt Fourier transform. The spectral representation of the Green’s fun&on is t,hcn obtained in Section III by taking the time l’ouricr transform. This forrn clearly shows that the function has poles at excitat,ion energies and residues which are rclatrd to state amplit,udes. The spectral reprcscnt’atjion can be put, in inkgral form by defining a spectral function. It is shown in Section IV that the spectral representation of t,hr many-time causal Green’s function can be reduced to t,he speckal rcprcsentation of the k-o-time Green’s function by appropriate integrations. In Section V the spectral representation of the two-time Green’s function is obtained directly and shown to be in agrccmentj
4.50
IiOBE
mit’h the one obtained from the many-time spect,ral reprcscntat~ion. Special cases of hhe t,wo-time spectral representation are examined and arc shown to bc essentially the same as those obtained by other workers (3, 10). II.
I~EFIPCITIO~
OF
THE
GRISEN’S
FUiYCTIOSS
The many-time causal Green’s function considered in this paper will bc defined for fermions, since by using commutation relations instead of ant,icommut’ation relat)ions it can be adapted to bosons. The most general type of Green’s functions that, do not, necessarily conserve particle number will be considered here. These Gwen’s functions arise in the treatment of the interactions brtwcen quasi-particles in a many fcrmion syst.em, since tht quasi-particles arc not con served (12). Tht particle conserving Gwen’s functions are just a special case of t,hese Green’s funct’ions in which the number of creat’ion and annihilation opcrators are cyunl. This many-time causal Green’s function is drfincd in coordinate and time space as r,,,(l,
2, . . . ) n; H. +
1, . . . ) N)
= i(T{$(l)#(Y)
. . . l+qn)$b+(n
+
1) . . . l)‘(N)))
(1)
where in this context j = (xj , cj , tj) , and Xj , aj = (=t) , and tj are the coordinate, spin (,up/down) , and time associated with parMe j, respcct,ively. The operators G and #+ are the annihilation and creation opcrat,ors for fermions in the Heisenberg pict,ure. The yuant#ity 1~is the number of annihilation operators and 171 is t,hc number of creation operat,ors, so ~1+ I)Z = N, the total number of operators in the Green’s funct#ion. The operator 1’ t’ime ordrrs the product of field operators in the Heiscnbrrg picture with a positive (negat,ivc) sign for an even (odd) permut,ation of the above order. The average in Ey. (I) is taken wit,h respect to t,hr ground state of t’he int,eract’ing particles. However, the dependence of the syst’rm on t’he temperat,ure can be obtained by t,aking thr averagr with respect t’o a grand canonical rnsemble (13). For simplicity only t,he zero t,emperat,ure case will br considered in t’his paper. The N-point, Green’s function of l&f. (1) is essentially the same as t’he kind considrrcd by Martin and Srhminger (1) and by Klein (14) when r~ = ~1 and t#he temperat,ure is absolute zero. The time independent field operators can be expanded in normal modes and substituted into Eq. (1). The lcourirr coefficient of t’his rxpansion is the Green’s function in momentum and time space (15) @,,(l,
2,
. . ) n; n + 1, . . . ) N)
where the numbers j = (kj , rj,
= i(T(a1
. . . WY:+1
. . . a‘v+J)
(2)
tj) are the momentum, spin, and time, rr-
SPECTRAL
351
REPRESEY-TATIOS
spwtively, sssoriated wit,h particle j. The operat.ors “j and ocj+ in Ey. (2) are t’he annihilation and creation operators for fermions in t,he Heisenberg pict,ure. The time> independent creat’ion and annihilation operators sat,isfy the usual fcrmion ant~icommut,ation relations. It will be c~onvenicnt. for obtaining a spectral representation of A,, to write it in twrns of the optrator A,. for r = 1, 2, 13, . . , N which is drfincd as
where t,he sum is taken over all permutations of 1, 2, . . , N and ( - 1) p is the sign of t#‘he permutation. The s&p function 0(r) is definr~d as il
if t _2 0
?0
ift
( 5)
e(t) = and arc used for thr time ordering are defined as ( ik - ili+I) = ( ti, in the Hciscnberg picAre. likfuat’ion spectral represent#at,ion, which will
<0
in Eq. (4). The argumrntjs of 0 in Eel. (4) t;,,,). The optrators A, in Ey. (4) are also (1) is in a form most suit’able for obtaining a hc done in next section.
The spc&al representation of t’hc Grwn’s function is the time lcourier t,ransform of the func%ion. In order to remove t’hc time dcpcndencc of t,hc operators in E(l. (-4) it is nwessary to make 11s;~ of the rompktt set of tigenst,atcs Is) of the t,otal Hamiltonian H which satisfy the Schriidinger equation Hlw) = I:‘,\jS)
(6)
I\-here I<,. is the eigen\-nlue of the atatc Is). The states Is) are assumed complete, so c
Is)(s~ = 1
(7)
where 1 is the unit operator. If Eys. (7) and (6) are subst.it,uted into Eq. (4) and use is made of t,he trans-
4.52
IiORE
formation T,k
=
t,,
TiN
=
tj,
-
lSk
f,,+,
(8) the result
The
is
oprratjors
in b;q. (9) are no longer
time
dependent,
and
wp = E,<- E.
(10)
is t#he excitation energy of the state is), since E. is the exact ground state energy of the interacting system. If the syst’em is parMe conserving, the number of particles in the state Is) must, he specified. The Vourier transform of Isq. (9) can now be taken. The general Fourier transform operator is 5(...)
= (2T)p”/m
If this operator is applied used, we obt,ain the spectral fun&ion
G,,,(l,
‘2, ..
-a
..’
/dtl
...
dt,exp
{i C
to Eq. (9) and the transformation representation of the many-time
>N) = ,i6(& + “.
+
AN)
.ddh, - wq) (slj~i,ls?)doi, .“C#J(X,,
(11)
Xjtj)[...)
jgN
+
F
(-ljpF
(ol-lih)
+ xi2 - us,) (S&4i3/S3) ...
+ &--I
of Eq. (8) is causal Green’s
-
w.SN-l)
( 12)
(SN--IIAi.,@)
where here j = (kj , gj , A,). The delta function gives over-all conservation of of frequency and occurs because the Green’s fun&ion is invariant under time translat’ion as can be seen from Eq. (9). The function 4(y) in Eq. (12) is the Fourier t’ransform [as defined in Eq. (I l)] of the step function f?(t) and is
(13) The factor exp( --iyt) in 4(y) of Eq. (13) occurs because the step function defined as one at zero, and the contribution from the contour at infinity must zero. The spectral representation in Eq. (12) is a complicated function of x2 ) . . . 1 XN , but it can be seen by using Eq. (13) that it has poles at the
0 is be X1 , ex-
citation energies for various sums of the frrcluencies. The residues do not have a simple significance, but do give information regarding the state amplitudes. The sprct,ral representat,ion of Eq. ( 12) can be rewritten in integral form by using a spectral function which involves a sum of delta func:t.ions so that when tht: integration is performed, 13q. (12) is again obtained. The many-time specstral rrprescnt#at,ion of Eq. ( 12) is somewhat cunlbersomc, so it will be of interest to reduce it to t,hr spectral represent,ation of the usual two-t,imc causal Green’s fun&on, which is done in the nest section. IV.
RI~:U~CTION
OF
THIC
MANY-TI?UZE: REPREWSTATIO~
TO
THIS
TWO-TIMIJ:
SPECTR.41,
Since the two-time Green’s function has been more useful in practice than the many-time causal Green’s function (,I) it is of interest to reduce the spectral reprcscntatIion of t)hr many-time fun&on to that of the two-time function. The general two-time causal Green’s function can be definc>d in terms of the manytime fur&on B,,, as s,,(k,
p; f, t’)
= /:
. . [ d!l . . . (1t.v GO a(t -
fj) (14)
. I,I, 6(f ‘-
t,)C\i,,,(l, 2, . . . , N)
The first, p argument)s in @,,, of l’:q. (2) are at a time t and the remaining arc at a time f’, where p is an arbitrary integer less than N = IL + 1~. The Fourirr transform [as defincld in Eq. (ll)] of 111~1.( 14) may bc taken to obt#ain (;,,,(k,
/3;, A, X’)
= /‘W . . . /” &I -+ci
. . . A,
6(X -
XI -
. -A,)
.6(X’ - x&9+1 - . . . -X,)G,,,(
1, 2,
. . , N)
where G,, m is the many-time causal Green’s function in momentum and frequcncy space which is given by the sprct,ral rcprcstntation of Eq. ( 12). Equation ( 15) shows just what, process in frequency space corresponds to setting the times c~lual in Eq. ( 14). If E(l. ( 12) is substituted into Eel. i 15) and t,he esprcssion int,egrat,cd with respect, to X’, it becomes C:,,,,,ik, /3, X) = ix where
I(I’,
t,hv int,rgrul
s) == /= “. -P.6(---x
P
(-I,‘c
(0/=2,,jsl)(sll~~~i,is~)
(s.~-&I~~~~OO)J(P,
S)
(l(i)
I( P, s) is
/ dhl . . . flX,G(X - Xl -
..
-x,) (17)
-x6+1
-
.'.
-x,)
rI .l
+ob,
+
"'
+
Lj
-
W?,)
-I-M
KOBE
The integral 1(1’, s) can be evaluated,
and is for I-’ = P’ for P = 1”’ - w,,_p) for all other permutations
(40 - WV,) I(P, where the permutations
sj = (4(-x lo
(18)
P’ and I’” are P’ =
(’ .
ip+l . i, 21 . . %@ ml . . mp rn,j,l ... mN >
(19)
and
(20) In Eqs. (19) and (20) t#hequantities I)L~are (1, 2, 3, .
, p - 1, or @
for 1 S j I /3 (21)
lllj = I /3+1,@+2;..,N-
1, orN
forfl+lsjsN
and each WL~ is different. Before substitut,ing Eq. ( 18) into Eq. ( 16) it, is convenirntj to define the two operators
and
where Pi is a permutation of 1, 2, . . , p, and /3 + 2, . . . , N.
P2 is a permutation of /3 + 1,
Therefore, if Eqs. (18)) ( 19), and ( 20) are used in Eq. ( Iti), and Ii:qs. (22) and (23) are also substituted, the result is G,m(k, P, X) = ix
(OlRljs)ddX - wJ(sl&lO)
+ ( - 1j(‘v-%c 9 (OlBz(s)$J( --A - w,) (sl&10)
(24)
which is the spectral represent,ation of the general two-time Green’s fun&ion. If Eq. (13) is substituted into Eq. (24)) it is clear that the function has poles at t,he excitation energies and residueswhich are state amplitudes. This spectral representation can be obtained directly from Eq. (14) as will be shown in t)he next section.
V. SPIXTRAI,
REPRESENTATIOX
OF
THE
TWO-TIMI:
GREES’S
FUSCT[OS
The spectral representation of t,he two-time Green’s fun&on is well known for c&ain cases (5, 10). For t’he sake of comparison, t,hc general t,wo-time spect,ral rcprescntat,ion of Eq. (24) will now be derived directly from F:q. ( l-1) . With the aid of Eqs. (2)) (22) and (23)) FIq. (14) becomes G,,,(k,
,!3, t, t’)
= i(OIR,(t)B,(t’)/0)8(t
-
t’) (‘5)
+
( - l)‘“-P’“i(OIB~(t’)Rl(t)
10)e(t’
-
1)
The Fourier transform of t,his equation gives Eq. (24). This method is, of wursc, the most, direct for obtaining the two-time spectral representation, but the mct,hod of the last, scct,ion gives some insight, into the relation bet,wccn the many-time and t,mo-time spectral representat,ions. Stvrral special cases of the two-time spectral rcprcsrntat,ion will nom he examined to show- that, essentially the same results are oht#ained here as have been obtained by others. In the wsc where /3 = ?z, Eqs. (22) and (33) can be writ ten as
by using the anticommut,ation relations for fermion operat,ors. stit,ut,ing P:qs. ( 26) and ( 13) into El. ( 24) we obtain Gnm(k, n, A) = -%
n!m!
.y (0 / a1 . . a,, 1 s)(s ( a:c.. a I ! X - w, + i0
Therefore,
sub-
aN+ IO) (27)
where the exponential t)erms in F:cl. ( IX) have been set equal to unitmy. If we flwthrr specialize to thr one part,irle Green’s function where n = w = Kq. (27) becomes
which is the same expression as used by Holst,erli (IO), alt,hough the factor 1,&r arises because the Fourier transform has brcn defined differently. VI.
1,
of
Co~CLI’sIos
The exact spectral representation of t)he many-time causal Green’s function obtained in this paper could lead t,o an increased importance of t,hese fun&ions in the many-body problem, since it, is now clear that t,he excit,ation energies of
t,he system, as well as additional information related to the state amplitudes, can bc obt,ained. l’ht> equations of motion satisfied by the many-time causal Green’s functions have been given by Martin and Schwinger (1) and by Klein (I,‘) for the purt8iclc conserving case. For t,he case of quasi-part#icles which are not wnsrrved, t.he equat’ions of motion will be given in a future paper (12). The equations of motion can be solved in some approximation to obtain npproximat,e Green’s funct,ions. The purpose of reducing the many-time spectral representation to the twotime speckal rcprcscnt,ation was to gain some insight into the rrlatSion between these rcprcscntat,ions. In part’icular, it is interestming to see what process in frecluemay space corresponds t,o sctt,ing tht times equal. The poles of t)he two-t’ime spectral rcpresentution also give excitation energies, but, only for those states which haw a nonzero state amplitude. The many-time spect,rnl representation has poles at, cssrntially all of’ the excit,at,ion energies. The residues of t,he twotime speckal representat)ion are state amplitudes, but it, is not clear just how the stat,e amplitudes would be obtained from the residues of the many-time spect,ral representations. AC~NOWI.EI)GJlESTH I would like to thank Professor W. R. Cheston
lZ~~:mvEn
: April
Professor for their
M. Bolsterli, 1)r. helpful discussions
d. A. ISvans, Dr. during the course
M. hluraskin, of t,his work.
and
30, 1962 RHFERENCES
1. P. C.MARTIN AND J. SCHWINGER,~~~J/S. fZel<. 1X,1342 11959). 2'. V. M. GALITSKII ATU A. H. M~c;oar,, Zhur. Eksptl. i. T~owt. Fiz. 34, 139 (1958) [translation: Sozrirt Phys.-JETP 7, 86: (1958)]. T. KATO. T. KORAYASHI, AND M. NAIWKI, Suppl. I',oy/~. Throrrl. Phys. iKyoto) No. 15, 3 (1960). A. Iirmx AND R. PRANGE, Whys. Rw. 112, 994 (1958). L). THOULESS, “The Quantum Mechanics of Many-Particle 8ystems.” rZcademic Press, Sew York, 1961. 3. V. I,. HON(WBRTEVICH ASD SH. M. KOGAK, .-lm. Phys. CSIT) 9, 125 (19tiO). V. I,. BONCH-BRLEVWH, Doklady .-1kad. .\-n,t/c SSSR 126, 539 11959) [t,ranslntion: Soviet Phys./Doklady 4, 596 (195911. SH. M. KOGAN, Doklady .4karl. -1.nuk SSSIZ 126, 546 (1959) [t.r:tnslation: Soviet Phys.mmD0kldy 4, 604 (1959jl. 4. I>. S. ZUEXRA~, TTspekhi Fiz. A\‘trrtk 71, 71 (1960) [translation: Soviet Phys.mmmC~spekhi 3, 320 (1960)]. 5. V. L. BONCH-BRI.EVICH, D~lild~~ ;Ikntl. .Yauk SSSR 129, 529 (1959) [translation: Soviet Phys:-Doklady 4, 1275 (1960)]. 6'. 13. LEIIAIANN, A'uozw cimento 191 11, 342 (1954). 7. I’. NAI\IBI-, Phys. Reo. 100, 394 (1955); ibid 101, 459 (1956); AA~uoz~o cinrento [lo] 6, 1054 (1957).
8.
9. 10. ff fd. IS.
14. 15.
lfi.
L. BOSCH-BRUEVICH, Zh,ur. Eksptl. i 7’eowt. Fiz. 31, 522 (1956) [tr:msl:tt~ion: Swirt Phys.-JETP 4, 456 (1957)]. I,. n. LANDAU, Zhur. h’ksptl. i l’rorrt. Fiz. 34, 262 (1958) [tran&t.ion: Sooiet Phyys-m JETP 7, 182 (1958)]. M. BOISTERIJ, Phys. IZw. Letters 4, 82 11960). iY. X, Booo~~ru~~ov ANI) $. I’. TYARLIKOV, Doklnrly .lknti. .Tn~k SSSfr’ 126, 53 (1959) [tjranslat,ion: Soviet Phys.m-Doklady 4, 580 11959)J. I). 1%. KOHE ANII W. B. CIIESTOS (to be published) See, for example, ref. 2, Thouless, Chap. VIIT, and ref. I. il. KLEIN, Theory of normal fermion systems. (To be published.) This t.ype of (ireen’s function was mentioned by iY. N. I~OGOLIL-BOV, 1). IV. Z~BAREV, AND I-I-. A. TSERKOVNIKOV, Zhur. Eksptl. i 7’eowt. Fiz. 39, 120 (1960) [translation: Sow’et Phys.-JE2’P 12, 88 (1961)]. This form was used by T’. P. (~ACIIOK, Zhw. Eksptl. i II’eoret. Fiz. 40, 879 (19Gt) [tr:lnHlntion: Swiet Phys.-JEl’P 13, 616 (1961)]. V.
OF
Spectral
PHYSICS:
19,
448-457 (1962)
Representation of the Function in Nonrelativistic Don-sun School
of Physics,
University
H.
Many-Time, Many-Body
Causal Green’s Theory*?
E(OBEf
of Minnesota,
Minneapolis
14,
Minnesota
The exact spectral representation of the many-time, causal Green’s function, which is the ground state average of the time ordered product of creation and annihilation operators in the Heisenberg picture, is obtained in this paper by taking the time Fourier transform of the function. This spectral representation shows that the function has poles at the excitation energies, and residues which are related to the st,at,e amplitudes. The usual two-time Green’s function can be written in terms of the many-time Green’s function, and the twotime spectral representat,ion obtained from the many-t,ime spectral representation by a process of integration. The two-time spectral represent,ation obtained in this manner agrees with the one obtained directly, but the derivation gives some insight into the relation between the two spectral representations. It is shown that in special cases the general two-time spectral representation obtained here is essentially the same as those obtained by others. I. INTROI>UCTION
The modern field t#heoretir technique of Green’s functions has become very important, in the last, few years in t#hc many-body problem ( 1, 2). I+?-omthe Green’s functions, which are averages of the time ordered product of field operators in the Heisenberg picture, the energy spectrum, information about, the state amplitudes, and even t#hct,hcrmodynamic proptr%ies of t’he system can be obt,ained (5, 2, 1). The Green’s fun&ions describe the creat,ion, propagation with consequent mutual and self-interactions, and subsequent annihilation of the particles. The t’wo-time causal, retarded, and advanced Green’s functions have been the most widely used C,$), since t’heir spectral reprcsentat8ions are well known. The spectral represent#ation of the two-time Green’s function, which can be obtained exactly, shows explicitly t’hat the poles of the funct’ion are t,he excitation energiesand the residuesare st#atcamplitudes. * This paper is based on part of a t,hesis submitted in parGal fulfillment of the requirements for the degree of I>octor of Philosophy at the University of Minnesota. t Supported by the U. S. .Stomic Energy Commission. $ Present address: Department of Physics and iZstronomy, The Ohio State University, Columbus 10, Ohio.
448
SPECTRAL
REPRESENTATIOS
448
The many-time causal Green’s functions have not been used as much in the many-body problem, because among other reasons the general spectral representations are not known (4). The spectral representat,ions of three-time t,hermal Green’s funct)ions have been studied by Bench-Bruevich (5) who also makes some general remarks about the many-t,ime functions. It, is the purpose of t#his paper to obt)ain the exact spectral representat8ion of the many-time, causal Green’s funct8ion and show its relation t,o t,he spect,ral representation of the twotime Green’s function. The spectral reprcsent,ation of Green’s functions or propagat,ors in rclat,ivistic field theory were first, obtained by Lchmann (6) and are thus known as Lehmann expansions. Nambu (7) has studied the structure of relativistic Green’s functions and obtained spectral representations which are parametric integrals over invariant energy denominators. However, he derived the representations on t.he basis of perturbation theory. III the nonrelnt,ivist,ic many-body problem, there is no Lorcntz invariance, so the relativistic speckal rcprcscnt,ations do not apply. For this case, exact spectral rcprcscntat8ions were first obtained by Bench-Brucvich (8) for the one and two particle Green’s funct,ions. He and Kogan (3) later extended this work to include general t,wo-t,ime temperat,urr dependent Green’s functions. Landau (9) has obtained a spectral representation of t)he single part,iclt, temperature dependent Green’s function, and used it, t,o derive a dispersion relation between the real and imaginary parts of the function. Bolsterli (10) has used t,he one and two particlc Green’s functions for systems of fcrmions with pairing forces to obt,ain the pairing energy, t,he chemical potential, and the excitation energies from an examination of their poles. The speckal representation of the single particle Green’s function has also been studied by Galit,skii and Migdal (2). A good review of twotime t,cmpcrat,ure dependent Green’s functions has been given by Zubarev (4) and he shows why the spectral representations of the retarded and advanced Green’s functions ( 11) are so useful. In Scct,ion II of this paper the many-time causal Green’s function is defined in coordinate and time space. Its Icourier coeficicnt is the Green’s function in momentum and time space, which can be rewritten in a more convenient form for taking t,hc t,imt Fourier transform. The spectral representation of the Green’s fun&on is t,hcn obtained in Section III by taking the time l’ouricr transform. This forrn clearly shows that the function has poles at excitat,ion energies and residues which are rclatrd to state amplit,udes. The spectral reprcscnt’atjion can be put, in inkgral form by defining a spectral function. It is shown in Section IV that the spectral representation of t,hr many-time causal Green’s function can be reduced to t,he speckal rcprcsentation of the k-o-time Green’s function by appropriate integrations. In Section V the spectral representation of the two-time Green’s function is obtained directly and shown to be in agrccmentj
4.50
IiOBE
mit’h the one obtained from the many-time spect,ral reprcscntat~ion. Special cases of hhe t,wo-time spectral representation are examined and arc shown to bc essentially the same as those obtained by other workers (3, 10). II.
I~EFIPCITIO~
OF
THE
GRISEN’S
FUiYCTIOSS
The many-time causal Green’s function considered in this paper will bc defined for fermions, since by using commutation relations instead of ant,icommut’ation relat)ions it can be adapted to bosons. The most general type of Green’s functions that, do not, necessarily conserve particle number will be considered here. These Gwen’s functions arise in the treatment of the interactions brtwcen quasi-particles in a many fcrmion syst.em, since tht quasi-particles arc not con served (12). Tht particle conserving Gwen’s functions are just a special case of t,hese Green’s funct’ions in which the number of creat’ion and annihilation opcrators are cyunl. This many-time causal Green’s function is drfincd in coordinate and time space as r,,,(l,
2, . . . ) n; H. +
1, . . . ) N)
= i(T{$(l)#(Y)
. . . l+qn)$b+(n
+
1) . . . l)‘(N)))
(1)
where in this context j = (xj , cj , tj) , and Xj , aj = (=t) , and tj are the coordinate, spin (,up/down) , and time associated with parMe j, respcct,ively. The operators G and #+ are the annihilation and creation opcrat,ors for fermions in the Heisenberg pict,ure. The yuant#ity 1~is the number of annihilation operators and 171 is t,hc number of creation operat,ors, so ~1+ I)Z = N, the total number of operators in the Green’s funct#ion. The operator 1’ t’ime ordrrs the product of field operators in the Heiscnbrrg picture with a positive (negat,ivc) sign for an even (odd) permut,ation of the above order. The average in Ey. (I) is taken wit,h respect to t,hr ground state of t’he int,eract’ing particles. However, the dependence of the syst’rm on t’he temperat,ure can be obtained by t,aking thr averagr with respect t’o a grand canonical rnsemble (13). For simplicity only t,he zero t,emperat,ure case will br considered in t’his paper. The N-point, Green’s function of l&f. (1) is essentially the same as t’he kind considrrcd by Martin and Srhminger (1) and by Klein (14) when r~ = ~1 and t#he temperat,ure is absolute zero. The time independent field operators can be expanded in normal modes and substituted into Eq. (1). The lcourirr coefficient of t’his rxpansion is the Green’s function in momentum and time space (15) @,,(l,
2,
. . ) n; n + 1, . . . ) N)
where the numbers j = (kj , rj,
= i(T(a1
. . . WY:+1
. . . a‘v+J)
(2)
tj) are the momentum, spin, and time, rr-
SPECTRAL
351
REPRESEY-TATIOS
spwtively, sssoriated wit,h particle j. The operat.ors “j and ocj+ in Ey. (2) are t’he annihilation and creation operators for fermions in t,he Heisenberg pict,ure. The time> independent creat’ion and annihilation operators sat,isfy the usual fcrmion ant~icommut,ation relations. It will be c~onvenicnt. for obtaining a spectral representation of A,, to write it in twrns of the optrator A,. for r = 1, 2, 13, . . , N which is drfincd as
where t,he sum is taken over all permutations of 1, 2, . . , N and ( - 1) p is the sign of t#‘he permutation. The s&p function 0(r) is definr~d as il
if t _2 0
?0
ift
( 5)
e(t) = and arc used for thr time ordering are defined as ( ik - ili+I) = ( ti, in the Hciscnberg picAre. likfuat’ion spectral represent#at,ion, which will
<0
in Eq. (4). The argumrntjs of 0 in Eel. (4) t;,,,). The optrators A, in Ey. (4) are also (1) is in a form most suit’able for obtaining a hc done in next section.
The spc&al representation of t’hc Grwn’s function is the time lcourier t,ransform of the func%ion. In order to remove t’hc time dcpcndencc of t,hc operators in E(l. (-4) it is nwessary to make 11s;~ of the rompktt set of tigenst,atcs Is) of the t,otal Hamiltonian H which satisfy the Schriidinger equation Hlw) = I:‘,\jS)
(6)
I\-here I<,. is the eigen\-nlue of the atatc Is). The states Is) are assumed complete, so c
Is)(s~ = 1
(7)
where 1 is the unit operator. If Eys. (7) and (6) are subst.it,uted into Eq. (4) and use is made of t,he trans-
4.52
IiORE
formation T,k
=
t,,
TiN
=
tj,
-
lSk
f,,+,
(8) the result
The
is
oprratjors
in b;q. (9) are no longer
time
dependent,
and
wp = E,<- E.
(10)
is t#he excitation energy of the state is), since E. is the exact ground state energy of the interacting system. If the syst’em is parMe conserving, the number of particles in the state Is) must, he specified. The Vourier transform of Isq. (9) can now be taken. The general Fourier transform operator is 5(...)
= (2T)p”/m
If this operator is applied used, we obt,ain the spectral fun&ion
G,,,(l,
‘2, ..
-a
..’
/dtl
...
dt,exp
{i C
to Eq. (9) and the transformation representation of the many-time
>N) = ,i6(& + “.
+
AN)
.ddh, - wq) (slj~i,ls?)doi, .“C#J(X,,
(11)
Xjtj)[...)
jgN
+
F
(-ljpF
(ol-lih)
+ xi2 - us,) (S&4i3/S3) ...
+ &--I
of Eq. (8) is causal Green’s
-
w.SN-l)
( 12)
(SN--IIAi.,@)
where here j = (kj , gj , A,). The delta function gives over-all conservation of of frequency and occurs because the Green’s fun&ion is invariant under time translat’ion as can be seen from Eq. (9). The function 4(y) in Eq. (12) is the Fourier t’ransform [as defined in Eq. (I l)] of the step function f?(t) and is
(13) The factor exp( --iyt) in 4(y) of Eq. (13) occurs because the step function defined as one at zero, and the contribution from the contour at infinity must zero. The spectral representation in Eq. (12) is a complicated function of x2 ) . . . 1 XN , but it can be seen by using Eq. (13) that it has poles at the
0 is be X1 , ex-
citation energies for various sums of the frrcluencies. The residues do not have a simple significance, but do give information regarding the state amplitudes. The sprct,ral representat,ion of Eq. ( 12) can be rewritten in integral form by using a spectral function which involves a sum of delta func:t.ions so that when tht: integration is performed, 13q. (12) is again obtained. The many-time specstral rrprescnt#at,ion of Eq. ( 12) is somewhat cunlbersomc, so it will be of interest to reduce it to t,hr spectral represent,ation of the usual two-t,imc causal Green’s fun&on, which is done in the nest section. IV.
RI~:U~CTION
OF
THIC
MANY-TI?UZE: REPREWSTATIO~
TO
THIS
TWO-TIMIJ:
SPECTR.41,
Since the two-time Green’s function has been more useful in practice than the many-time causal Green’s function (,I) it is of interest to reduce the spectral reprcscntatIion of t)hr many-time fun&on to that of the two-time function. The general two-time causal Green’s function can be definc>d in terms of the manytime fur&on B,,, as s,,(k,
p; f, t’)
= /:
. . [ d!l . . . (1t.v GO a(t -
fj) (14)
. I,I, 6(f ‘-
t,)C\i,,,(l, 2, . . . , N)
The first, p argument)s in @,,, of l’:q. (2) are at a time t and the remaining arc at a time f’, where p is an arbitrary integer less than N = IL + 1~. The Fourirr transform [as defincld in Eq. (ll)] of 111~1.( 14) may bc taken to obt#ain (;,,,(k,
/3;, A, X’)
= /‘W . . . /” &I -+ci
. . . A,
6(X -
XI -
. -A,)
.6(X’ - x&9+1 - . . . -X,)G,,,(
1, 2,
. . , N)
where G,, m is the many-time causal Green’s function in momentum and frequcncy space which is given by the sprct,ral rcprcstntation of Eq. ( 12). Equation ( 15) shows just what, process in frequency space corresponds to setting the times c~lual in Eq. ( 14). If E(l. ( 12) is substituted into Eel. i 15) and t,he esprcssion int,egrat,cd with respect, to X’, it becomes C:,,,,,ik, /3, X) = ix where
I(I’,
t,hv int,rgrul
s) == /= “. -P.6(---x
P
(-I,‘c
(0/=2,,jsl)(sll~~~i,is~)
(s.~-&I~~~~OO)J(P,
S)
(l(i)
I( P, s) is
/ dhl . . . flX,G(X - Xl -
..
-x,) (17)
-x6+1
-
.'.
-x,)
rI .l
+ob,
+
"'
+
Lj
-
W?,)
-I-M
KOBE
The integral 1(1’, s) can be evaluated,
and is for I-’ = P’ for P = 1”’ - w,,_p) for all other permutations
(40 - WV,) I(P, where the permutations
sj = (4(-x lo
(18)
P’ and I’” are P’ =
(’ .
ip+l . i, 21 . . %@ ml . . mp rn,j,l ... mN >
(19)
and
(20) In Eqs. (19) and (20) t#hequantities I)L~are (1, 2, 3, .
, p - 1, or @
for 1 S j I /3 (21)
lllj = I /3+1,@+2;..,N-
1, orN
forfl+lsjsN
and each WL~ is different. Before substitut,ing Eq. ( 18) into Eq. ( 16) it, is convenirntj to define the two operators
and
where Pi is a permutation of 1, 2, . . , p, and /3 + 2, . . . , N.
P2 is a permutation of /3 + 1,
Therefore, if Eqs. (18)) ( 19), and ( 20) are used in Eq. ( Iti), and Ii:qs. (22) and (23) are also substituted, the result is G,m(k, P, X) = ix
(OlRljs)ddX - wJ(sl&lO)
+ ( - 1j(‘v-%c 9 (OlBz(s)$J( --A - w,) (sl&10)
(24)
which is the spectral represent,ation of the general two-time Green’s fun&ion. If Eq. (13) is substituted into Eq. (24)) it is clear that the function has poles at t,he excitation energies and residueswhich are state amplitudes. This spectral representation can be obtained directly from Eq. (14) as will be shown in t)he next section.
V. SPIXTRAI,
REPRESENTATIOX
OF
THE
TWO-TIMI:
GREES’S
FUSCT[OS
The spectral representation of t,he two-time Green’s fun&on is well known for c&ain cases (5, 10). For t’he sake of comparison, t,hc general t,wo-time spect,ral rcprescntat,ion of Eq. (24) will now be derived directly from F:q. ( l-1) . With the aid of Eqs. (2)) (22) and (23)) FIq. (14) becomes G,,,(k,
,!3, t, t’)
= i(OIR,(t)B,(t’)/0)8(t
-
t’) (‘5)
+
( - l)‘“-P’“i(OIB~(t’)Rl(t)
10)e(t’
-
1)
The Fourier transform of t,his equation gives Eq. (24). This method is, of wursc, the most, direct for obtaining the two-time spectral representation, but the mct,hod of the last, scct,ion gives some insight, into the relation bet,wccn the many-time and t,mo-time spectral representat,ions. Stvrral special cases of the two-time spectral rcprcsrntat,ion will nom he examined to show- that, essentially the same results are oht#ained here as have been obtained by others. In the wsc where /3 = ?z, Eqs. (22) and (33) can be writ ten as
by using the anticommut,ation relations for fermion operat,ors. stit,ut,ing P:qs. ( 26) and ( 13) into El. ( 24) we obtain Gnm(k, n, A) = -%
n!m!
.y (0 / a1 . . a,, 1 s)(s ( a:c.. a I ! X - w, + i0
Therefore,
sub-
aN+ IO) (27)
where the exponential t)erms in F:cl. ( IX) have been set equal to unitmy. If we flwthrr specialize to thr one part,irle Green’s function where n = w = Kq. (27) becomes
which is the same expression as used by Holst,erli (IO), alt,hough the factor 1,&r arises because the Fourier transform has brcn defined differently. VI.
1,
of
Co~CLI’sIos
The exact spectral representation of t)he many-time causal Green’s function obtained in this paper could lead t,o an increased importance of t,hese fun&ions in the many-body problem, since it, is now clear that t,he excit,ation energies of
t,he system, as well as additional information related to the state amplitudes, can bc obt,ained. l’ht> equations of motion satisfied by the many-time causal Green’s functions have been given by Martin and Schwinger (1) and by Klein (I,‘) for the purt8iclc conserving case. For t,he case of quasi-part#icles which are not wnsrrved, t.he equat’ions of motion will be given in a future paper (12). The equations of motion can be solved in some approximation to obtain npproximat,e Green’s funct,ions. The purpose of reducing the many-time spectral representation to the twotime speckal rcprcscnt,ation was to gain some insight into the rrlatSion between these rcprcscntat,ions. In part’icular, it is interestming to see what process in frecluemay space corresponds t,o sctt,ing tht times equal. The poles of t)he two-t’ime spectral rcpresentution also give excitation energies, but, only for those states which haw a nonzero state amplitude. The many-time spect,rnl representation has poles at, cssrntially all of’ the excit,at,ion energies. The residues of t,he twotime speckal representat)ion are state amplitudes, but it, is not clear just how the stat,e amplitudes would be obtained from the residues of the many-time spect,ral representations. AC~NOWI.EI)GJlESTH I would like to thank Professor W. R. Cheston
lZ~~:mvEn
: April
Professor for their
M. Bolsterli, 1)r. helpful discussions
d. A. ISvans, Dr. during the course
M. hluraskin, of t,his work.
and
30, 1962 RHFERENCES
1. P. C.MARTIN AND J. SCHWINGER,~~~J/S. fZel<. 1X,1342 11959). 2'. V. M. GALITSKII ATU A. H. M~c;oar,, Zhur. Eksptl. i. T~owt. Fiz. 34, 139 (1958) [translation: Sozrirt Phys.-JETP 7, 86: (1958)]. T. KATO. T. KORAYASHI, AND M. NAIWKI, Suppl. I',oy/~. Throrrl. Phys. iKyoto) No. 15, 3 (1960). A. Iirmx AND R. PRANGE, Whys. Rw. 112, 994 (1958). L). THOULESS, “The Quantum Mechanics of Many-Particle 8ystems.” rZcademic Press, Sew York, 1961. 3. V. I,. HON(WBRTEVICH ASD SH. M. KOGAK, .-lm. Phys. CSIT) 9, 125 (19tiO). V. I,. BONCH-BRLEVWH, Doklady .-1kad. .\-n,t/c SSSR 126, 539 11959) [t,ranslntion: Soviet Phys./Doklady 4, 596 (195911. SH. M. KOGAN, Doklady .4karl. -1.nuk SSSIZ 126, 546 (1959) [t.r:tnslation: Soviet Phys.mmD0kldy 4, 604 (1959jl. 4. I>. S. ZUEXRA~, TTspekhi Fiz. A\‘trrtk 71, 71 (1960) [translation: Soviet Phys.mmmC~spekhi 3, 320 (1960)]. 5. V. L. BONCH-BRI.EVICH, D~lild~~ ;Ikntl. .Yauk SSSR 129, 529 (1959) [translation: Soviet Phys:-Doklady 4, 1275 (1960)]. 6'. 13. LEIIAIANN, A'uozw cimento 191 11, 342 (1954). 7. I’. NAI\IBI-, Phys. Reo. 100, 394 (1955); ibid 101, 459 (1956); AA~uoz~o cinrento [lo] 6, 1054 (1957).
8.
9. 10. ff fd. IS.
14. 15.
lfi.
L. BOSCH-BRUEVICH, Zh,ur. Eksptl. i 7’eowt. Fiz. 31, 522 (1956) [tr:msl:tt~ion: Swirt Phys.-JETP 4, 456 (1957)]. I,. n. LANDAU, Zhur. h’ksptl. i l’rorrt. Fiz. 34, 262 (1958) [tran&t.ion: Sooiet Phyys-m JETP 7, 182 (1958)]. M. BOISTERIJ, Phys. IZw. Letters 4, 82 11960). iY. X, Booo~~ru~~ov ANI) $. I’. TYARLIKOV, Doklnrly .lknti. .Tn~k SSSfr’ 126, 53 (1959) [tjranslat,ion: Soviet Phys.m-Doklady 4, 580 11959)J. I). 1%. KOHE ANII W. B. CIIESTOS (to be published) See, for example, ref. 2, Thouless, Chap. VIIT, and ref. I. il. KLEIN, Theory of normal fermion systems. (To be published.) This t.ype of (ireen’s function was mentioned by iY. N. I~OGOLIL-BOV, 1). IV. Z~BAREV, AND I-I-. A. TSERKOVNIKOV, Zhur. Eksptl. i 7’eowt. Fiz. 39, 120 (1960) [translation: Sow’et Phys.-JE2’P 12, 88 (1961)]. This form was used by T’. P. (~ACIIOK, Zhw. Eksptl. i II’eoret. Fiz. 40, 879 (19Gt) [tr:lnHlntion: Swiet Phys.-JEl’P 13, 616 (1961)]. V.