Fermions in spherical field theory Dean Lee12 University of Massachusetts Amherst, MA 01003 9 We derivethesphericalfieldformalismforfermions. Wefindthatthespher- 9 ical field method is free from certain difficulties which complicate lattice 9 calculations, such as fermion doubling, missing axial anomalies, and compu- 1 tational problems regarding internal fermion loops. n a [PACS numbers: 11.10.Kk, 11.15Tk] J 7 1 Overview 2 v 7 Spherical field theory is a new non-perturbative method for studying quan- 4 1 tum field theory. It was introduced in [1] and was used to describe the 2 interactions of scalar boson fields. In this paper we show how to extend the 1 8 spherical field method to fermionic systems. 9 Thecentralideaofsphericalfieldtheoryistotreatad-dimensionalsystem / h as a set of coupled one-dimensional systems. This is done by expanding field t - configurations of the functional integral in terms of spherical partial waves. p e Regarding each partial wave as a distinct field in a new one-dimensional the- h ory, we interpret the functional integral as a time-evolution equation, where : v the radial distance in the original theory serves as the time parameter. For a i X purely bosonic system the time-evolution equation corresponds with a mul- r tidimensional partial differential equation. In the case of a purely fermionic a system, we find that the time evolution is described by a system of first-order ordinary differential equations. In future work we will study mixed systems with both bosons and fermions which are described by coupled partial dif- ferential equations. Unlike lattice methods, spherical field theory yields an expansion which, at any order, corresponds with a continuous system. It is therefore able to avoid problems associated with discrete approximation methods.3 There is no doubling of fermion states, and we find the correct axial anomaly. Fur- 1Supported by the National Science Foundation under Grant 5-22968 2email: [email protected] 3The author credits Robert Shrock for pointing this out. 1 thermore internal fermion loops present no special computational difficulties and is included in the dynamics of the time-evolution equation. Detailed examples of such calculations will be presented in a forthcoming paper. We anticipate that spherical field methods will be useful in the study of non- perturbative fermionic systems, especially chiral fermions and phenomena related to fermion loop processes. Theorganizationofthispaperisasfollows. Webeginwithabrief descrip- tion of Grassmanian path integrals and the fermionic analogof the Feynman- Kac formula. We then generate the spherical expansion for free fermion the- ory in two Euclidean dimensions with sources and derive the spherical field time-evolution operator and generating functional. By functional differen- tiation with respect to the sources, we obtain the spherical field formalism for general interacting theories. Next we check that spherical field theory produces the correct axial anomaly. We then show how to write the time- evolutionequationasamatrixsystemandcomment ontheutilityofspherical field methods in studying fermionic systems. Although our analysis is done in two dimensions, the extension to higher dimensions is straightforward. 2 Grassmanian path integrals ¯ Let ψ (t) and ψ (t) be Grassman-valued functions where i,j = 1,···,N. Let i j ¯ ¯ V(ψ ,ψ ,t) be a polynomial in ψ and ψ , ordered such that all ψ’s are placed i j i j ¯ on the right and all ψ’s are placed on the left. In [2] it is shown that tF Tr T exp − dtV(a+,a ,t) (1) i −j (cid:20) (cid:26) ZtI (cid:27)(cid:21) ∝ dΨdΨ¯ DψDψ¯exp − tF dt Nk=1ψ¯kddψtk , ψ(tI)= ψ(tF)=Ψ +V(ψ¯,ψ ,t) Z Zψ¯(tI)=−ψ¯(tF)=Ψ¯ ( ZtI " P i j #) − where the trace is performed over the space spanned by vectors of the form |s ···s i s = 0,1;···s = 0,1; (2) 1 N 1 N and hs ···s |s ···s i = δ ···δ (3) ′1 ′N 1 N s′1s1 s′NsN a+|s ···s ···s i = (−1)s1+ +si−1δ |s ···1···s i i 1 i N ··· si,0 1 N a |s ···s ···s i = (−1)s1+ +si−1δ |s ···0···s i. −i 1 i N ··· si,1 1 N 2 This is the fermionic version of the Feynman-Kac formula. A more recent derivationusing fermioniccoherent statescanbefoundin[3]and[4]. Thean- tiperiodic boundary conditions imposed at t and t follow as a consequence I F of computing the trace. We note that these are in fact special conditions. More general boundaryconstraints produce ambiguities which depend onthe specific discrete approximation used to obtain the continuum limit. It is not clear that such antiperiodic boundary conditions can be general- ized in a coordinate-independent manner for functional integrals over higher dimensional regions. The rigorous theory of Grassmanian functional inte- gration has not developed to the point where we can answer such questions. Nevertheless functional integrationis aconvenient method forderiving useful field-theoretic results, althoughin a somewhat heuristic fashion. Inthis anal- ysis we use the functional integral to deduce the spherical field formalism for fermions. Although we will be careless with regard to boundary conditions, in the end we explicitly check that the spherical field method produces the correct generating functional for free field theory. By functional differentia- tion with respect to the external sources, we conclude that the spherical field formalism is valid for general interacting theories. 3 Spherical fermions Let us consider Euclidean field theory in two dimensions. We will use both cartesian and polar coordinates, ~t = (tcosθ,tsinθ) = (x,y). (4) In Euclidean space the gamma matrices satisfy γi,γj = −2δij, (5) n o and we choose the representation ~γ = i~σ. (6) Let us start by constructing the spherical field Hamiltonian. We first decom- pose the fermion fields, ψ = ψ↑(~t) = √12πψn↑(t)einθ , (7) ψ (~t) 1 ψ (t)einθ " ↓ # n=0, 1, " √2π n↓ # X± ··· 3 ψ¯ = ψ¯ (~t) ψ¯ (~t) = 1 ψ¯ (t)einθ 1 ψ¯ (t)einθ . (8) ↑ ↓ √2π n↑ √2π n↓ h i n=0X,±1,···h i Using 0 ∂ −i ∂ 0 e iθ ∂ − i ∂ ~σ ·∇~ = ∂x ∂y = − ∂t t∂θ , (9) " ∂ +i ∂ 0 #  eiθ ∂ + i ∂ (cid:16) 0 (cid:17)  ∂x ∂y ∂t t∂θ  (cid:16) (cid:17)  we have ~σ ·∇~ √12πψn↑(t)einθ = 1  (cid:18)∂∂ψtn↓ + ntψn↓(cid:19)ei(n−1)θ . (10) " √12πψn↓(t)einθ # √2π  ∂∂ψtn↑ − ntψn↑ ei(n+1)θ   (cid:18) (cid:19)    The Euclidean action for free field theory with external sources, η and η¯, is S = −i dθdtt ψ¯(i~γ ·∇~ −m)ψ +ψ¯η +η¯ψ . (11) Z (cid:16) (cid:17) In terms of partial waves,4 −ψ¯↑n+1 ∂∂ψtn↓ + ntψn↓ −ψ¯↓n 1 ∂∂ψtn↑ − ntψn↑ S = −i dtt  − (cid:18) ¯ (cid:19) −¯ − (cid:18) (cid:19) . −m ψ↑nψn↑ +ψ↓nψn↓ Z n=0X,±1,··· +ψ¯↑nηn↑ +(cid:16)ψ¯↓−nηn↓ +η¯↑n−ψn↑ +(cid:17)η¯↓nψn↓   − − − − (12) The generating functional is therefore Dψi Dψ¯i exp ∞dtt G , (13)  n n  n Z Yi,n Z0 n=0X,±1,···    where G is defined as   n −ψ¯↑n ∂ψ∂n↓t+1 + n+t1ψn↓+1 −ψ¯↓n 1 ∂∂ψtn↑ − ntψn↑ (14) − (cid:18) (cid:19) − − (cid:18) (cid:19) ¯ ¯ ¯ ¯ −m ψ↑nψn↑ +ψ↓n 1ψn↓+1 +ψ↑nηn↑ +ψ↓n 1ηn↓+1 +η¯↑nψn↑ +η¯↓n 1ψn↓+1. − − − − − − − − − (cid:16) (cid:17) If we now define ψ¯i = tψ¯i, the generating functional is, up to an overall n′ n constant, Dψi Dψ¯i exp ∞dt G , (15)  n n′  ′n Z Yi,n Z0 n=0X,±1,···  4We expand η andη¯into partial waves in the same manner as ψ and ψ¯.   4 where G is ′n −ψ¯↑′n ∂ψ∂n↓t+1 + n+t1ψn↓+1 −ψ¯↓′n 1 ∂∂ψtn↑ − ntψn↑ (16) − (cid:18) (cid:19) − − (cid:18) (cid:19) ¯ ¯ ¯ ¯ −m ψ↑′nψn↑ +ψ↓′n 1ψn↓+1 +ψ↑′nηn↑ +ψ↓′n 1ηn↓+1 +tη¯↑nψn↑ +tη¯↓n 1ψn↓+1. − − − − − − − − − (cid:16) (cid:17) Our goal is to find an equivalent expression for (15), in analogy with the Feynman-Kac formula (1). We start by defining a linear vector space. For each finite subset n = 0,±1,±2,··· S ⊂ si (17) n i =↓,↑ ( (cid:12) ) (cid:12) (cid:12) we assign a vector |Si satisfying(cid:12) the following orthogonality and normaliza- (cid:12) tion conditions, 0 if S 6= S ′ hS |Si = (18) ′ 1 if S = S . ( ′ For later convenience we define a lexicographic order, namely, si < si′ (19) n n′ if and only if n < n (20) ′ or n = n, i =↓ , and i =↑ . ′ ′ Let Σ be the linear space spanned by all such vectors |Si. Let us define operators ai+ and ai by the following relations, n n− 0 if si ∈/ S ai |Si = n (21) n− (−1)#|S −sii if si ∈ S, ( n n 0 if si ∈ S ai+|Si = n n (−1)#|S ∪sii if si ∈/ S, ( n n where # is the number of elements in S which are less than si. Comparing n (15) with (1), we make the correspondences5 ψn↑,ψ¯↓′n 1 ↔ a↑n−,a↑n+ (22) − − ψn↓+1,ψ¯↑′n ↔ a↓n−+1,a↓n++1 − 5We denote the conjugateofain− asain+. Althoughain− correspondswith apartialwave withorbitalangularmomentum n,we note thatai+ correspondswith apartialwavewith n orbital angular momentum −n−1 or −n+1, depending on i. 5 and define Tr Texp ∞dt Hε(η¯,η,t) − 0 n Z[η¯,η] = lim (cid:20) (cid:26) n=0,±1,··· (cid:27)(cid:21), (23) ε→0 Tr Texp −R0∞dt P Hnε(0,0,t) (cid:20) (cid:26) n=0,±1,··· (cid:27)(cid:21) R P where Hnε = n+t1a↓n++1a↓n−+1 − nta↑n+a↑n− +(m+ε)a↓n++1a↑n− +(m+ε)a↑n+a↓n−+1(24) −a↓n++1ηn↑ −a↑n+ηn↓+1 −tη¯↑na↑n− −tη¯↓n 1a↓n−+1. − − − The traces in (23) are performed over the space Σ. In practise it is not necessary to explicitly compute these traces, since only the ground state projection contributes. For m = 0, however, the ground state is degenerate and the ε → 0 prescription picks out the correct ground state. We will refer to (23) as the fermionic spherical field ansatz. For notational ease we will suppress the ε terms. We now show that Z[η¯,η] is the correct generating functional for free field theory. We note that Σ can be decomposed as a tensor product space with the identification |Si ↔ S ∩{s↑n,s↓n+1} . (25) n=0O,±1,···(cid:12)(cid:12) E Since H (η¯,η,t) acts upon only the (cid:12)nth component in the tensor product, n we can write Z[η¯,η] as a product Z[η¯,η] = Zn[η¯↑n,η¯↓n 1,ηn↑,ηn↓+1], (26) − − − n=0, 1, Y± ··· where Zn[η¯−↑n,η¯−↓n−1,ηn↑,ηn↓+1] = TTrr[[TTeexxpp{{−−R00∞∞ddttHHnn((η0¯,,0η,,tt))}}]]. (27) In (27) the trace is performed over the four-dimRensional space spanned by the vectors |∅i, {s↓n+1} , {s↑n} , {s↑n,s↓n+1} . (28) (cid:12) E (cid:12) E (cid:12) E In Appendix 1 we derive(cid:12)the result,(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) t η (t ) Z = expn=0X,±1,···Z dt1dt2h t1η¯−↑n(t1) t1η¯−↓n−1(t1) iMn(t1,t2)" t2η2n↓+n↑1(2t2) #, (29)   6 where the matrix M (t ,t ) is defined as6 n 1 2 mK (|m|t )I (|m|t ) |m|K (|m|t )I (|m|t ) θ(t −t ) n 1 n 2 n 1 n+1 2 (30) 1 2 |m|K (|m|t )I (|m|t ) mK (|m|t )I (|m|t ) " n+1 1 n 2 n+1 1 n+1 2 # mI (|m|t )K (|m|t ) −|m|I (|m|t )K (|m|t ) +θ(t −t ) n 1 n 2 n 1 n+1 2 2 1 −|m|I (|m|t )K (|m|t ) mI (|m|t )K (|m|t ) " n+1 1 n 2 n+1 1 n+1 2 # for m 6= 0, and θ( n 1)tn θ(n+1)tn 0 − −2 1 0 2 1 tn+1 tn+1 θ(t −t ) 2 −θ(t −t ) 2 (31) 1 2  θ(n+1)tn  2 1  θ( n 1)tn  2 2 0 − −2 2 0 tn+1 tn+1  1   1      for m = 0. Let us now compare these results with the known results for free field theory. The two-point free field correlator is ∆ij(~t) = d2~k ~k~γ+mei~k~t = d2~k i~k~σ+mei~k~t (32) (2π)2 −~k2·+m2 · (2π)2 −~k2+· m2 · Z Z = d2~k ~σ ~+mei~k~t. (2π)2 −~k2·+∇m2 · Z ~ Integrating over k, we find ∆ij(~t) = 1 [mK (|m|t)+|m|K (|m|t)·(σxcosθ+σysinθ)] (33) 2π 0 1 mK (|m|t) |m|e iθK (|m|t) = 1 0 − 1 2π |m|eiθK (|m|t) mK (|m|t) " 1 0 # for m 6= 0. When m → 0 we find 0 e iθ1 ∆ij(~t) = 1 − t . (34) 2π eiθ1 0 " # t From (33) and (34) it is straightforward to recover the spherical correlation functions in (30) and (31). We conclude that the fermionic spherical field ansatz produces the correct generating functional for free fermions. These correlation functions are part of the spherical Feynman rules for fermions. For future reference we have written these in a more convenient format in Appendix2. Byfunctionaldifferentiationwithrespect tothesourcesη andη¯, we conclude that the spherical field formalism is valid for general interacting theories. 6Ii and Ki are the ith order modified Bessel functions of the first and second kinds respectively. 7 4 Axial anomaly Wenowshow thatthespherical fieldformalismyields thecorrectformforthe axial anomaly. We consider free massless fermions, again in two Euclidean dimensions. Let us define Sµν(~t) = 0 Vµ(~t)Aν(0) 0 , (35) E D (cid:12) (cid:12) E (cid:12) (cid:12) where Vµ and Aµ are the vector an(cid:12)d axial vecto(cid:12)r currents, and the subscript E is intended as a reminder that our Euclidean correlationfunction isdefined as the analytic continuation of the corresponding time-ordered function in Minkowski space.7 We note that 0 Vµ(~t)∂ Aν(0) 0 = −∂~t 0 Vµ(~t)Aν(0) 0 = −∂ Sµν. (36) ν ν ν E E D (cid:12) (cid:12) E D (cid:12) (cid:12) E (cid:12) (cid:12) (cid:12) (cid:12) Theone-l(cid:12)oopprocessco(cid:12)rresponding wit(cid:12)hSµν carrie(cid:12)salogarithmicdivergence proportional to εµν, and the regulated value of Sµν will depend on our def- initions of Vµ and Aµ as operator products. In our discussion here we will remove this ambiguity by considering the symmetric combination Sµν+Sνµ. Let us define vector and axial vector currents, Vµ(~t) = ψ¯(~t)γµψ(~t) = iψ¯(~t)σµψ(~t) (37) Aµ(~t) = ψ¯(~t)γµγ ψ(~t) = iψ¯(~t)σµσzψ(~t). (38) 5 Expanding the currents in terms of partial waves, we have Vµ(~t) = 21π ei(k−n)θ vµ ψ¯↑n(t)ψk↓(t)+vµ ψ¯↓n(t)ψk↑(t) (39) ↑↓ − ↓↑ − k,n=X0,±1,··· h i Aµ(~t) = 21π ei(k−n)θ aµ ψ¯↑n(t)ψk↓(t)+aµ ψ¯↓n(t)ψk↑(t) , (40) ↑↓ − ↓↑ − k,n=X0,±1,··· h i where v1 = i, v1 = i, v2 = 1, v2 = −1 (41) ↑↓ ↓↑ ↑↓ ↓↑ a1 = −i, a1 = i, a2 = −1, a2 = −1. (42) ↑↓ ↓↑ ↑↓ ↓↑ 7Euclidean correlation functions can also be defined without analytic continuation. However this involves new complications which are described in [5]. 8 From (39) and (40), we have Sµν(~t) = 1 0 e2iθvµ ψ¯1↑(t)ψ1↓(t) aν ψ¯0↑(0)ψ0↓(0) 0 . (2π)2 * (cid:12)(cid:12)(cid:12)" +e−2iθv↑↓µ↓↑ψ¯−↓1(t)ψ−↑1(t) #" +a↑↓ν↓↑ψ¯0↓(0)ψ0↑(0) #(cid:12)(cid:12)(cid:12) +E(43) (cid:12) (cid:12) (cid:12) (cid:12) Recalling the correlator results from the previous section, we have Sµν(~t) = 1 e2iθvµ aν +e 2iθvµ aν , (44) (2π)2t2 − ↑↓ ↑↓ ↓↑ ↓↑ (cid:16) (cid:17) and so −∂ Sµν −∂ Sνµ = − 4i δµ1 ∂ 2xy − ∂ x2 y2 (45) ν ν (2π)2 ∂x (x2+y2)2 ∂y (x2+−y2)2 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) + 4i δµ2 ∂ x2 y2 + ∂ 2xy . (2π)2 ∂x (x2+−y2)2 ∂y (x2+y2)2 (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) Integrating withany smoothtest functionwerecognize these terms asderiva- tives of the Dirac delta distribution, −∂ Sµν −∂ Sνµ = − 4i δµ1 −π∂ δ(~t) + 4i δµ2 −π∂ δ(~t) (46) ν ν (2π)2 2 (2π)2 1 = iεµν∂ δ(~t)h. i h i π ν We conclude that d2teip~~t 0 Vµ(~t)∂ Aν(0) 0 − 0 ∂ Vν(~t)Aµ(0) 0 = 1εµνp . (47) · ν E ν E π ν Z hD (cid:12) (cid:12) E D (cid:12) (cid:12) E i (cid:12) (cid:12) (cid:12) (cid:12) If we now choos(cid:12)e to maintain(cid:12)conservati(cid:12)on of the vecto(cid:12)r current, we have d2teip~~t 0 Vµ(~t)∂ Aν(0) 0 = 1εµνp , (48) · ν E π ν Z D (cid:12) (cid:12) E (cid:12) (cid:12) which is the desired result for(cid:12) the axial ano(cid:12)maly (see [6]). This should not be surprising. With ∂ Aν placed at the origin, the calculation we have done ν is the same as that of standard field theory in position space. For the case when ∂ Aν is not at the origin, our spherical field calculation generates an ν expansion of Sµν(~t−~t) = 0 Vµ(~t)Aν(~t) 0 (49) ′ ′ E D (cid:12) (cid:12) E in terms of sums of spherical waves.(cid:12) The parti(cid:12)al sums of this expansion (cid:12) (cid:12) converge pointwise (except at~t =~t) to the result (44), and we again get the ′ correct axial anomaly. 9 5 Matrix representation Spherical fermion fields can be studied using ordinary matrices. Unlike the spherical bosonic system which corresponds with a multidimensional partial differential equation, the spherical fermionic system corresponds with a set of coupled first-order ordinary differential equations. We illustrate some basic methods here using the free fermion system. Let us define the following column vectors 0 0 0 1 0 0 1 0  0  = |∅i,  1  = {s↓n+1} ,  0  = {s↑n} ,  0  = {s↑n,s↓n+1} .     (cid:12) E   (cid:12) E   (cid:12) E  1   0  (cid:12)  0  (cid:12)  0  (cid:12)     (cid:12)   (cid:12)   (cid:12)         (50) We now write the Hamiltonian and creation and annihilation operators as matrices, 1 0 0 0 t 0 n m 0 H (0,0,t) =  −t  (51) n 0 m n+1 0  t   0 0 0 0      and 0 −1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 a↓n++1 =  0 0 0 1  a↓n−+1 =  0 0 0 0  (52)      0 0 0 0   0 0 1 0          0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 a + =   a =  . ↑n 0 0 0 0 ↑n− 1 0 0 0      0 0 0 0   0 1 0 0          As an illustrative example we calculate the correlation function ¯ h0|ψ1↓(t)ψ0↑(0)|0iE (53) for the case m 6= 0. Let t2 U(t ,t ) = T exp − dtH (0,0,t) . (54) 2 1 0 (cid:26) Zt1 (cid:27) 10