aa r X i v : . [ m a t h - ph ] O c t Stochastic quantization of massive fermions
A.N. EfremovCPHT, Ecole Polytechnique, CNRS, Universit´e Paris-Saclay,Route de Saclay, 91128 Palaiseau, [email protected] 8, 2018
Abstract
We consider a general solution of the Langevin equation describingmassive fermions to an appropriate boundary problem. Assumingexistence of such solution we show that its correlators coincide withthe Schwinger functions of corresponding Euclidean Quantum FieldTheory.
In this work we make an effort to fill a gap in literature and to establish amathematically sound connection between the Langevin equation and mas-sive fermionic models in the quantum field theory. A motivation for this anal-ysis is our desire to extend the methods developed recently by A.Kupiainen [1]and M.Hairer [2] to non-abelian gauge theories, Yukawa and Gross–Neveumodels [3, 4]. Compared with a functional integral approach stochasticPDE’s present a clear simplification and allow us to consider a much widerclass of models. We hope that the problem of criticality which puts severelimitation on the dimension of time-space can be solved using Wilson’s non-perturbative renormalization group [5]. A common way to perform stochasticquantization is of course the Fokker–Planck equation, see Parisi and Wu [7].Unfortunately a fermionic field is a function in infinite-dimensional Grass-mann algebra [6] and despite its random or stochastic character, is not astochastic process as we understand it in the probability theory. Although1he Fokker–Planck approach suggested in [8] for fermions is simple, intuitiveand better suited for an initial reading we should keep in mind that the wholedescription remains at a formal level only. On the other hand the functionalintegral approach makes our construction for fermions mathematically mean-ingful. Rather than formally derive the Fokker–Planck equation for fermionswe follow the functional integral approach proposed by J. Zinn-Justin for thescalar φ model [9].The first important object is the Euclidean action. Without any lossof generality we consider Dirac spinors in two dimensions. The 2-pointsSchwinger function for Dirac spinors can be obtained from the correspond-ing Wightman function on an appropriately chosen subspace by the Wickrotation [10], i.e. by the mapping x
7→ − ix , W ( x ) = Z d p (2 π ) − i ˆ /p + mp + m − iǫ e ipx , (1) S ( x ) = Z d p (2 π ) − i/p + mp + m e ipx . (2)Here ˆ /p = − p ˆ γ + p γ , ˆ γ = − iγ , /p = p k γ k , { γ i , γ j } = 2 δ ij , γ + i = γ i .The matrices ˆ γ i , γ i correspond to Minkowski and Euclidean space-time re-spectively. We choose the representation for the matrices γ i such that theycoincide with the Pauli matrices: γ = (cid:18) , , (cid:19) γ = (cid:18) , − ii, (cid:19) γ = (cid:18) , , − (cid:19) (3)Hence γ T = γ , γ = − iγ γ . It is easy to see that the function in (2) isa fundamental solution to a non-homogeneous Dirac equation in Euclideanspace-time, i.e. ( /∂ + m ) S ( x ) = δ ( x ) . (4)As an example we consider the Yukawa model [3]. The model involves ascalar field φ and a fermion field ψ , L = Z d x ¯ ψ ( /∂ + m ) ψ + g : ¯ ψψ : φ − φ ∆ φ + 12 M φ , (5)where ψ, ¯ ψ : R → Λ and φ : R → R are respectively independent spinorand scalar fields. Here by Λ we denote an infinite-dimensional Grassmannalgebra. Since the expectation h ¯ ψψ i turns out to be divergent we should2ntroduce a regularization to properly define the Wick product : ¯ ψψ :. Bydefinition the Wick product includes the ordinary product ¯ ψψ and a coun-terterm c which becomes singular when we remove the regularization, i.e.: ¯ ψψ : = ¯ ψψ − c . The matrices γ i transform as vectors under the rotation uγ i u − where u = e iγ α . Since γ is hermitian u is unitary. The quantities¯ ψψ and ¯ ψ /∂ψ are scalars, i.e. invariant with respect to the action ¯ ψ → ¯ ψu + and ψ → uψ , in appendix B we state other properties of the action.The second important object is the Langevin equation: ∂ t φ = − δLδφ + ξ = ∆ φ − M φ − g : ¯ ψψ : + ξ, (6) ∂ t ψ = − δLδ ¯ ψ + η = − ( /∂ + m ) ψ − gψφ + η, (7) ∂ t ¯ ψ = δLδψ + ¯ η = ( /∂ T − m ) ¯ ψ − g ¯ ψφ + ¯ η, (8)where ξ , η , ¯ η denote the corresponding Gaussian white noise. Having asolution of these equations one can calculate different correlators. Below weshow that such correlators coincide with the Schwinger functions which weobtain using the corresponding generating functional. First let us write the fundamental solution of the linear equations correspond-ing to the retarded Green functions( ∂ t + /∂ + m ) G = δ ( t, x ) , (9)( ∂ t − /∂ T + m ) ¯ G = δ ( t, x ) , (10)( ∂ t − ∆ + M ) P = δ ( t, x ) . (11)These functions vanish ∀ t <
0, i.e. P ( t, x ) = 0, G ( t, x ) = 0 and ¯ G ( t, x ) = 0. G ( t, x ) = 1 i Z dwd p (2 π ) w − im − /p ( w − im ) − p e iwt + ipx , (12)¯ G ( t, x ) = 1 i Z dwd p (2 π ) w − im + /p T ( w − im ) − p e iwt + ipx , (13) P ( t, x ) = 1 i Z dwd p (2 π ) w − i ( p + M ) e iwt + ipx . (14)3ere ¯ G = γ Gγ . We can write equations (6)-(8) in the following form ∂ t Ψ + ˜ D Ψ + V (Ψ) − Ξ = 0 , (15)whereΞ = η ¯ ηξ , Ψ = ψ ¯ ψφ , ˜ D = /∂ + m − /∂ T + m − ∆ + M , V (Ψ) = g ψφ ¯ ψφ : ¯ ψψ : . (16)Here Ψ( t, x ) is a distribution in ( a, b ) × R which satisfies an initial valueproblem Ψ( t , x ) = Ψ ( x ) for equation (15), t ∈ ( a, b ) ⊂ R . Then we putequation (15) into an integral formΨ = G [+ ∞ ,t ] ( − V + Ξ) + e − ˜ D ( t − t ) Ψ , G = G G
00 0 P (17)( G [+ ∞ ,t ] f )( t, x ) = ∞ Z t ds Z d z G ( t − s, x − z ) f ( s, z ) . (18)This result follows immediately from the definition of the retarded Greenfunction G , i.e. ( ∂ t + ˜ D ) G = δ ( t, x ). We also assume that a solution for theinitial value problem exists for an infinite time interval, t = −∞ . Conse-quently we can identify the time interval ( a, t ) with R . When t = −∞ ,since m > t, x ) should solve a fix point problemΨ = G ( − V + Ξ) . (19)Here and everywhere in the text below G f = G [+ ∞ , −∞ ] f . Under the aboveassumption and an appropriate regularity of the potential V the choice ofmodel, the Yukawa model in two dimensions in our case, is not important inthe coming lemmas whose proofs hold in general, so we try to keep generalnotations as much as possible.Following the great success in the construction of the massive scalar modelusing methods of functional integral [12] K.Osterwalder and R.Schrader con-structed the Euclidean theory of free fermions [11]. To put it simply, givenan action L which involves fermions one defines a generating functional Z ft for connected Schwinger functions using an integral over infinite dimensional4rassmann algebra [6] along with the usual functional integral for bosons.Denoting these fields by ˜Ψ = ( ˜ ψ, ˜¯ ψ, ˜ φ ) we have Z ft ( ˜ K ) = Z D ˜Ψ e − L + ˜Ψ Q ˜ K , Q = − , ˜ K = ˜ k ˜¯ k ˜ j . (20)Here ˜ k, ˜ ψ : R → Λ and ˜ j, ˜ φ : R → R in two dimensions. To shorten thenotation we use ˜Ψ Q ˜ K instead of R d x ˜Ψ( x ) Q ˜ K ( x ). The composite field ˜Ψdoes not depend on the stochastic time. Given a fixed time T we could define˜Ψ( x ) = Ψ( T, x ) but it is not necessary here. We use a tilde to emphasize thata variable or an operator is explicitly independent of the stochastic time. Forconvenience we introduce the notation˜ E (Ψ) = ˜ D Ψ + V (Ψ) . (21)Since the elements of Grassmann algebra are anti-commuting we will distin-guish the left δ l and right derivatives δ r . Lemma 1
The partition function Z ft satisfies the following equation (cid:18) ˜ E ( Q T δδ l ˜ K ) − ˜ K (cid:19) Z ft ( ˜ K ) = 0 . (22) Proof ˜ KZ ft ( ˜ K ) = Z D Ψ e − L (cid:16) Q T δe Ψ Q ˜ K δ l Ψ (cid:17) = Z D Ψ (cid:16) − Q T δe − L δ l Ψ (cid:17) e Ψ Q ˜ K (23)= Q T δLδ l Ψ (cid:12)(cid:12)(cid:12) Ψ= Q T δ l ˜ K Z ft ( ˜ K ) = ˜ E ( Q T δ l ˜ K ) Z ft ( ˜ K ) . (24) (cid:4) Let Ψ be a solution of the fix point problem given in (19). Define the followinga generating functional Z : Z ( K ) = Z D Ξ e − Ξ Q Ξ+ KQ Ψ , K = k ¯ kj . (25)5ere k, ψ : R → Λ and j, φ : R → R in two dimensions. Wewant to show that at some finite time t = T ∈ R the generating func-tional Z ( K ) satisfies equation (22). Since Z (0) = Z ft (0) = 1 it will implythat Z ( K ) = Z ft ( ˜ K ), i.e. the correlators obtained from (25) at time T , i.e. h Ψ( T, x ) ... Ψ( T, x n ) i , coincide with the Schwinger functions of (20). Lemma 2
Let Ψ be a solution of the fix point problem stated in (19) . Fur-thermore, let ˜ Z ( ˜ K ) = Z ( K ) where K ( t, x ) = δ ( t − T ) ˜ K ( x ) . The partitionfunction ˜ Z satisfies the equation (cid:18) ˜ E ( Q T δδ l ˜ K ) − ˜ K (cid:19) ˜ Z ( ˜ K ) = 0 . (26) Proof
For the expectation of the noise using (25) we obtain h Ξ i K = Q T δ Ψ T δ l Ξ QKZ ( K ) , δ Ψ T δ l Ξ = G T (cid:16) δ V T δ l Ψ G T (cid:17) − . (27)Furthermore ∂ t Ψ = (cid:16) G δ V δ r Ψ (cid:17) − ∂ t G Ξ . (28)Since Ψ satisfies the Langevin equation, i.e. ∂ t Ψ + ˜ E (Ψ) − Ξ = 0 we have h(cid:16) G δ V δ r Ψ (cid:17) − ∂ t G − i Ξ + ˜ E (Ψ) = 0 . (29)Calculating the expectation of (29) and using (27) we find (cid:16) ˜ E + h(cid:16) G δ V δ r Ψ (cid:17) − ∂ t G − i Q T G T (cid:16) δ V T δ l Ψ G T (cid:17) − QK (cid:17) Z ( K ) = 0 . (30)We restrict our interest to the correlators at t = T , i.e. K = δ ( t − T ) ˜ K ( x ).Since G ( t, x ) = 0 for t < t → h G T (cid:16) δ V T δ l Ψ G T (cid:17) − i t,x = lim t → G T ( t, x ) = δ ( x ) . (31)Defining U r = δ V δ r Ψ , ˜ R = (cid:16) GU r (cid:17) − ∂ t G Q T G T (cid:16) U T l G T (cid:17) − , (32)6e rewrite equation (30) in the form(˜ E − ˜ K + ˜ RQ ˜ K ) Z ( K ) = 0 (33)It remains to prove that equations (33) and (26) are equivalent. One canshow that for the Green functions the following holds ∂ t G t − τ,x − z Q T G Tτ − t ,z − y = Q T G Tt − t ,x − y − G t − t ,x − y Q T . (34)This quantity vanishes whenever t = t = T . Expanding the inverse opera-tors appearing in (32) in power series over U we obtain for a fixed order m the following expression m X n =0 ( GU r ) n ( Q T G T − G Q T )( U T l G T ) m − n = m X n =1 ( GU r ) n Q T G T ( U T l G T ) m − n − m − X n =0 ( GU r ) n G Q T ( U T l G T ) m − n + Q T G T ( U T l G T ) m − ( GU r ) m G Q T . (35)Here the last two terms vanish if the time argument on the both ends is thesame. Thus for m > − m − X n =0 ( GU r ) n G [ Q T U T l − U r Q T ] G T ( U T l G T ) m − − n . (36)After summing up all orders m we obtain the final expression for ˜ R at equaltime on the both ends˜ R = (cid:16) GU r (cid:17) − G [ Q T U T l − U r Q T ] G T (cid:16) U T l G T (cid:17) − . (37)The quantity Q T U T l − U r Q T vanishes, see appendix A. Consequently thedifference between equations (33) and (26) vanishes. (cid:4) I thank the Institute for Theoretical Physics at the University of Leipzig,Germany for the financial support. 7 U r and U T l Since in a general situation the potential V includes all required countertermswhich are fine-tuned to cancel singularities we still use here the same notationas if we have the whole action L . Furthermore all derivatives below are leftderivatives. U r = − δ Lδ ¯ ψ i δψ j − δ Lδ ¯ ψ i δ ¯ ψ j δ Lδ ¯ ψ i δφ j δ Lδψ i δψ j δ Lδψ i δ ¯ ψ j − δ Lδψ i δφ j − δ Lδφ i δψ j − δ Lδφ i δ ¯ ψ j δ Lδφ i δφ j = u u u u u u u u u (38) U T l = δ Lδψ i δ ¯ ψ j − δ Lδψ i δψ j δ Lδψ i δφ j δ Lδ ¯ ψ i δ ¯ ψ j − δ Lδ ¯ ψ i δψ j δ Lδ ¯ ψ i δφ j δ Lδφ i δ ¯ ψ j − δ Lδφ i δψ j δ Lδφ i δφ j = u − u − u T − u u − u T u T u T u (39)The equation Q T U T l = U r Q T implies u = u T and u = − u T . B Symmetries of the action
We define as usual the time and parity reversal operators T : (cid:0) x , x (cid:1) (cid:0) − x , x (cid:1) , P : (cid:0) x , x (cid:1) (cid:0) x , − x (cid:1) , (40)and corresponding transformations for the spinor ψP : ψ ( x ) γ ψ ( P x ) , T : ψ ( x ) γ ψ ( T x ) , C : ψ ( x ) γ ¯ ψ ( x ) , (41)where T is anti-linear, i.e. T αψT − = α ∗ T ψT − . The action L , see (5), isinvariant under P . Invariance under CT can be obtained if one simultaneousmakes inversion of the sign of mψ ( x ) γ ¯ ψ ( T x ) , ¯ ψ ( x ) ψ ( T x ) γ , m
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