We derive an infinite hierarchy of integral equations for the Green functions of a many-particle system. This set of equations forms the basis of a unified approach.
International Journal of Theoretical Physics, Vol. 23, No. 2, 1984
New Approach to Perturbation Theory of Many-Particle Systems C. J. Lambert and W. E. Hagston Physics Department, The Universi(v. Hull. HU6 7RX, England Received March 25. 1981 We derive an infinite hierarchy of integral equations for the Green functions of a many-particle system. This set of equations forms the basis of a unified approach to the perturbation theory of many boson and many fermion systems and avoids the introduction of the adiabatic hypothesis. It is demonstrated how a well-known ground state perturbation theory of a system of interacting fermions is obtained without introducing disconnected diagrams. It is shown that the formalism allows a self-consistent determination of the condensate Green function of a condensed Bose system and a derivation of the Beliaev, Hugenholtz, and Pines result for the single-particle k ~ 0 Green function is given. A new self-consistent equation for the k = 0 Green function is solved to yield the well-known selfenergy relation ~11 - Y02 =/.t which plays the role of a self-consistency condition on the theory.
1. I N T R O D U C T I O N A n exact, first-principles, m a n y - b o d y p e r t u r b a t i o n theory d e s c r i p t i o n of the p r o p e r t i e s of liquid 4He is still lacking. T o e x a m i n e the p o s s i b i l i t y of c o n d e n s a t i o n into the k = 0 m o d e , it is n a t u r a l to focus a t t e n t i o n on the single-particle c o n d e n s a t e (i.e., k = 0) G r e e n f u n c t i o n g o ( t - t'). D e n o t i n g the n o r m a l i z e d i n t e r a c t i n g g r o u n d state b y Iq~N) a n d a H e i s e n b e r g picture a n n i h i l a t i o n o p e r a t o r for the k = 0 m o d e b y a0,,(t), this q u a n t i t y is d e f i n e d (Beliaev, 1958)
iG o (t - t ' ) = ((BH(tl)'"BH(tr-I)[aHk(tr)
' nH( tr)] - S
• Bn(tr+ ,)...Be(t;))O(t - tr) - i f/,dt'T( X~I (t') Be(t I )...Be(tl))e i~k'' + e-~lC'le"~
(A15)
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Lambert and Hagston
This equation is only valid for t t > t". To remove this restriction, consider repeating the analysis for some other value of t. Denoting this value of t by T and subtracting the result from equation (A15) yields
e -Eltlei'~ktT(al4k(t)Blf(tl)...Bn(tt)) -- e - Elrlei'~krT{ ank( t )BH( tl)...Bn( tt) ) I = E s r - l e - elt'lei~~ r=l
• ( B n ( t , ) . . . B H ( t r- ,)[aHk(/r) , Btt(tr) ] _sBH(tr q- 1)...BH(t/) ) • (O(t -- tr)-- O ( T - tr) ) - i f ~ d t ' T ( X n ( t ' ) B n ( t , ) . . . B n ( t , ) ) e
'~k'' (A16)
This equation is valid for arbitrary t and T. Furthermore, it is valid for arbitrary (ti). To see this, substitute equation (17) into the first term on the right-hand side. Each term in the resulting equation contains l operators B(ti). Hence for arbitrary (ti) the operators B(t~)...B(tt) can be simultaneously reordered on both sides and any factors S p resulting from P permutations cancel exactly. The derivation of equation (15) is completed after replacing Bn(ti)... Bn(tt) by ~n(2)...q~n(m)q, tn(l')...~k~(n') and noting that
APPENDIX B: DERIVATION OF EQUATIONS (46) AND (47) In the thermodynamic limit, the first term on the right-hand side of equation (43) vanishes and a combination of this equation with (44) yields
no e -i~(t,- ti) = no e -i~(t,- t~) o(0,/~) II
(B1)
where o(k, ~) is the Fourier transform of the self-energy o. Hence, from (B1) o(0,/x) =/~
(B2)
a(0,/~) = Y,,, (0,/~)- Xo2 (0,/~)
(B3)
We proceed to demonstrate that
Perturbation
Theoryof Many-Particle Systems
123
Our analysis is essentially based on a trivial modification of the arguments employed by Abrikosov et al. (1963) to derive equation (47). The average of U~(t0, - to) with respect to the vacuum of the operators ~b' and ~b't is given by (Beliaev, 1958)
U(to,-to)=T[e
or]
(B4)
where 3' is a functional of a 0 and at0. 3' is a sum of vacuum loops (Beliaev, 1958) and in general, eacb contribution to 3' contains m pairs of interaction t , t , ). Let "7 be the function picture operators ao(t,)...ao(t,~)ao(q)...ao(t,, obtained by replacing such strings of operators in 3' by (36). (In this way, we replace operators a0(ti)[at0(t~) ] by dotted lines entering (leaving) the points q[t~].) Then 7 will be a c-number function which may be written :
(BS)
where 7,, is the sum of all vacuum loops containing m pairs of dotted lines. A careful consideration of the analysis which led to equations (40)-(43) reveals the following: (i) o(0,/z) is obtained by first removing a single arbitrarily chosen ingoing dotted line from 7 and then removing in all possible ways an outgoing dotted line. Thus m
o(0, ~) : ~ n--~o3,--"
(B6)
(ii) Y~,,(0,/~) is obtained by removing an ingoing and outgoing dotted line from 7 in all possible ways: m 2
(By)
(iii) Finally, Y.o2(0,/~) is obtained by removing a pair of outgoing dotted lines from 7 in all possible ways: Y.o2(0,/z ) = y~ m ( m m
I) T,,,
(B8)
17 0
A combination of equations (B2), (B6) to (B8) yields the desired result.
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Lambert and Hagston ACKNOWLEDGMENTS
Dr. Lambert wishes to thank the Science Research Council for the provision of a research studentship and the Irish Department of Education for the provision of a research fellowship.
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