EQUATIONS OF MOTION LUCA GUIDO MOLINARI

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1 EQUATIONS OF MOTION LUCA GUIDO MOLINARI 1. Equation of motion of destuction opeatos Conside a system of bosons o femions descibed by a Hamiltonian H = H 1 + H 2, whee H 1 and H 2 ae espectively the one and two paticle opeatos H 1 = h ab c ac b, H 2 = 1 (1) v abcd c 2 ac b c dc c ab abcd An abitay one-paticle basis is used, with canonical opeatos c a and c a. The matix elements ae h ab = a h b and v abcd = ab v cd = v badc (exchange symmety). The time evolution of a destuction opeato, c (t) = e iht/ c e iht/, solves the equation of motion i d dt c (t) = e iht/ [c, H]e iht/. By means of the commutatos (2) [c, c a c b] = δ a c b (3) [c, c ac b c dc c ] = (δ a c b ± δ bc a)c d c c one evaluates [c, H 1 ] = b h bc b and [c, H 2 ] = 1 2 bcd (v bcd ± v bcd )c b c dc c ; indices c, d ae exchanged in the second tem, with c c c d = ±c d c c. Then [ψ, H 2 ] = 1 2 bcd (v bcd+v bdc )c b c dc c. The two matix elements ae equal because of exchange symmety. The final expession is: (4) [c, H] = b h b c b + bcd v bcd c b c dc c An immediate consequence is the useful opeato identity: (5) c [c, H] = H 1 + 2H 2 Anothe consequence is the equation of motion of a destuction opeato: (6) i d dt c (t) = b h b c b (t) + bcd v bcd (c b c dc c )(t) 2. The gound state enegy Eq.(5) yields an expession fo the total enegy, due to Galitskii and Migdal. The expectation value on the exact gound state is H 1 + 2H 2 = i c (t) d dt c d (t) = i lim t t + dt c (t )c (t) Date: nov

2 2 LUCA GUIDO MOLINARI Since t > t a T-odeing can be intoduced in the inne poduct. This allows to exchange the opeatos and obtain: (7) H 1 + 2H 2 = d G (t, t + ) dt whee the one-paticle Geen function, in a geneic basis, is: ig (t, t ) = Tc (t)c (t ) = θ(t t ) c (t)c (t ) ± θ(t t) c (t )c (t) The equation povides the expectation value of the inteaction in tems of the onepaticle Geen function. The total enegy is E gs = H 1 + H 2 : (8) E gs = ± i 2 lim [ ] d i δ ab t t + dt + h ab G ba (t, t ) ab Execise. Show that, in pesence of space-time tanslation invaiance and spin independent inteaction, the fomula (8) simplifies to E gs = ±(2s + 1)V i d 3 kdω 2 (2π) 4 [ ω + ǫ( k)]g( k, ω)e iωη whee V is the volume. Evaluate the integals fo the ideal electon gas. Let us evaluate 3. Equation of motion of the popagato i t G (t, t ) = δ(t t ) c (t)c (t ) c (t )c (t) + T dc (t) c dt (t ) = δ(t t )δ i b h b Tc b (t)c (t ) i bcd v bcd T (c b c dc c )(t)c (t ) To emove the paenthesis, one has to fix the ambiguity of equal time opeatos in the T poduct. The ode of opeatos is peseved if the infinitesimal time shifts ae intoduced: T(c b c dc c )(t)c (t ) = Tc b (t++ )c d (t + )c c (t)c (t ). This allows to pemute them feely unde the T odeing (but the plus must be left in place to estoe the oiginal ode): = Tc c (t)c d (t + )c (t )c b (t++ ). The two-paticle Geen function is intoduced: (9) i 2 G abcd (t a, t b, t c, t d ) = Tc a (t a )c b (t b )c d (t d)c c (t c) The equation of motion of the popagato is the fist of a hieachy of equations fist obtained by Matin and Schwinge which involve highe ode Geen functions: (i t ) (10) δ b h b G b (t t ) = δ δ(t t ) b +i v bcd G cdb (t, t +, t ++, t ) bcd

3 EQUATIONS OF MOTION 3 In position epesentation, and fo spin independent inteactions, the equation is: (i t ) (11) h( x) G mm ( xt, x t ) = δ mm δ 3 ( x x )δ(t t ) +i d 3 y v( x, y)g m m ( xt, yt+, yt ++, x t ) In 4-dimensional notation, with U 0 (x, x ) = v( x, x )δ(t t ): (i t ) (12) h( x) G mm (x, x ) = δ mm δ 4 (x x ) +i d 4 y U 0 (x, y)g m m (x, y+, y ++, x ) If the two-paticle inteaction is absent, the equation of motion does not involve highe ode functions. Let us pause fo a while on Geen functions of noninteacting paticles. 4. Independent paticles Fo independent paticles the Geen function is a genealized function that solves the equation (13) [i δ ab t h ab ]G bc (t, t ) = δ ab δ(t t ) Its vey usefulness appeas in the solution of the inhomogeneous equation [i δ ab t h ab ]f b (t) = g a (t) with unknown f a (t) and assigned souce g a (t). The geneal solution is the sum of the geneal solution of the homogeneus equation and a paticula solution that can be geneated though the Geen function: f a (t) = fa 0 (t) + 1 dt G ab (t, t )g(t ). It must be noted that (13) does not have a unique solution (we have the feedom to add a solution of the homogeneous poblem). This is bette seen in fequency space, whee (13) is: [ ωδ ab h ab ]G bc (ω) = δ ab. The solution is well defined fo all eal ω if a pesciption is given on how to avoid the poles and cuts, i.e. the eal spectum of the single paticle Hamiltonian h. Thee ae vaious choices. The most useful ones ae the etaded and the time-odeed Geen functions. In the etaded Geen function the spectum of h is shifted by an infinitesimal imaginay pat to the lowe imaginay half-plane: (14) G R ab (ω) =: (ω h/ + iη) 1 ab = n a n n b ω ω n + iη whee h n = ω n n, and we neglect the continuum. In passing we note that the imaginay pat of the diagonal matix elements in the position basis give the local density of states: (15) 1 π ImGR ( x, x; ω) =: n x n 2 δ(ω ω n ) The tace (and this is basis-independent) is the density of states of the Hamiltonian. Since the etaded Geen function is analytic in the uppe half plane (povided that

4 4 LUCA GUIDO MOLINARI it vanishes fo lage ω ), its Fouie tansfom to the time vaiables is is zeo fo t > t: + ig R ab(t, t dω ) =i 2π GR ab(ω)e iω(t t ) (16) (17) (18) = θ(t t ) e iωn(t t ) a n n b n = θ(t t ) a U(t, t ) b This featue is of geat impotance in physics as it expesses causality: the paticula solution fa R (t) = dt G R ab(t, t )g b (t ) only depends on the values g(t ) at t < t. In a many body system, the etaded Geen function is the expectation value of the commutato (bosons) o anticommutato (femions) at unequal times (the definition holds also fo inteacting systems): [ ] (19) ig R ab(t, t ) = gs c a (t), c b (t ) gs In the time odeed Geen function a efeence Femi fequency divides the spectum into a potion that gains a positive imaginay pat and anothe that gains a negative imaginay coection: (20) G T ab(ω) =: n a n n b ω ω n + iη sign(ω ω F ) The Fouie tansfom to time vaiables is (in position basis) (21) ig T ( x, t; x, t ) = n e iωn(t t ) x n n x [θ(t t )θ(ω ω F ) θ(t t)θ(ω F ω)] fo t > t the popagation involves enegy states above the Femi fequency (fo femions: paticle excitations), fo t < t it involves (fo femions) hole excitations. The expession coesponds to the geneal definition (also fo inteacting paticles) (22) ig T ab(t, t ) = gs Tc a (t)c b (t ) gs Note that ig T (t, t )±ig R (t, t ) is an odinay function that solves the homogeneous equation. Execise. Evaluate the etaded Geen function fo fee paticles (the esult does not depend on statistics) ig R ( x, t; x, t ) = θ(t t d 3 k ) (2π) 3 ei k ( x x ) iω k (t t ) [ ] 3/2 = θ(t t m ) exp[ 2π (t t i m x x 2 ] ) 2 t t.

5 EQUATIONS OF MOTION 5 By means of the unpetubed Geen function, the equation of motion fo the one paticle Geen function can be witten in integal fom: G mm (x, x ) = G 0 mm (x, x ) + i d 4 y d 4 y (23) G 0 m (x, y)u0 (y, y )G m (y, y +, y ++, x ) The equation G = G 0 + G 0 U 0 G 4 can be compaed with the Dyson equation fo the pope self-enegy, G = G 0 + G 0 Σ G, to expess the self enegy in tems of G 4 (epeated indices ae summed o integated): (24) Σ m(x, y)g m (y, x ) = i U0 (x, y)g m m (x, y+, y ++, x ) 5. Hatee Fock appoximation The two-paticle Geen function admits a decomposition into connected components: (25) G abcd (t a, t b, t c, t d ) = G ac (t a, t c )G bd (t b, t d ) ± G ad (t a, t d )G bc (t b, t c ) + G c abcd(t a, t b, t c, t d ) One of the seveal equivalent ways to pefom the Hatee Fock appoximation is to neglect completely the connected pat of the two paticle Geen function, meaning that the two paticles evolve independently. This tuncates the Matin-Schwinge hieachy of equations to the lowest level. The equation of motion fo the one paticle Geen function becomes closed (and quadatic): (i ddt h( x) ) G mm ( xt, x t ) = δ mm δ 3 ( x x )δ(t t ) ± i m d 3 yv( x, y) [ G m ( yt, x t )G m( xt, yt + ) ± G ( yt, yt+ )G mm ( xt, x t ) ] Note that ±i m G ( yt, yt+ ) = n( y). One obtains the Hatee inteaction: U H ( x) = d 3 y v( x, y)n( y). The equation of motion is ( i d ) dt h( x) U H( x) G mm ( xt, x t ) = δ mm δ 3 ( x x )δ(t t ) +i d 3 y v( x, y) G m ( yt, x t )G m( xt, yt + ) In ω space: ( ω h( x) U H ( x)) G mm ( x, x, ω) = δ mm δ 3 ( x x ) +i dω d 3 y v( x, y) G m ( y, x, ω) 2π G m( x, y, ω )e iηω To solve the equation we assume a spectal epesentation typical of independent paticles (26) G mm ( x, x, ω) = a u a ( x, m)u a ( x, m ) ω ω a + iηsign(ω a ω F )

6 6 LUCA GUIDO MOLINARI with othonomal functions u a and eal fequencies ω a. Inset the epesentation in the equation fo G, multiply by u a ( x, m ) and integate in x ad sum on m, Because of othogonality: u a ( x, m) [ ω h( x) U H ( x)] ω ω a ± iη = u a( x, m) +i d 3 u a ( y, ) dω y v( x, y) ω ω a ± iη 2π G m( x, y, ω )e iηω Hee U H is evaluated with n( y) = u b ( x, m) 2 θ(ω F ω b ). b m The integal in ω is evaluated by esidues and gives: i b u b( x, m)u b ( y, ) θ(ω F ω b ). Next, the limit ω ω a is taken, and the system of Hatee-Fock equations is obtained: (27) [h( x) + U H ( x)] u a ( x, m) b, θ(ω F ω b )u b ( x, m) d 3 y v( x, y) u a ( y, )u b ( y, ) = ω a u a ( x, m) If the Hatee Fock appoximation is done in eq.(24), one eads the HF appoximation fo the self enegy: Σ mm (x, x ) = ±i δ mm δ 4(x x )U 0 (x, y)g (y, y+ ) + i G mm (x, x + )U 0 (x, x ) We obtained anothe chaacteization of Hatee Fock appoximation: the HF selfenegy is povided by the two self enegy gaphs of fist ode with the self-consistent G eplacing G 0.