[x,t) by the Langevin equation; where r}(x, t) is a stochastic distribution and |^ acts as a damping termJ3) So we interpret the problem of quantizing the classical field in (d)-dimensions as a problem of non-equilibrium statistical mechanics in (d+l)-dimensions. The hope is that we may choose the distribution T]{x,t) so that in the large "time" limit {{x,t)(/)(y,t)) approaches the quantized correlation function. PART I: The Langevin Method of Stochastic Quantization 5 Choose T)(x,t) to be a Markovian (white noise) distribution, so that (rj(xit)) = 0 (V(x,t)r}(y,t')) = 26d(x - y)6(t - t'). (1.6) and, in general, We can represent {F[r}]) in terms of two point averages by making use of an arbitrary source term J(x,t)'} 1 6n {rji(xU*i) ...rjn(x„.tn)) = Z 6J(xi,ti)...J(xn,tn) J c%]e* f dd'dti2(',t)+J(',th(x,t) > (1.8) where z = J' d[n\e^di'il^l\ With this formalism, we find that {Vi • • • ^ n) =0 if n is odd = (tyi^ ) • • • (^ n-i'/n) + permutations if n is even, where rji = ^(x,-,/,). We want to use definition (1.7) to evaluate an arbitrary correlation of the form [Fl^jj]) where $ n is the solution to the Langevin equation (1.5). Once the action is specified, we have all the information necessary to perform the calculation. PART I: The Langevin Method of Stochastic Quantization However, we have yet to establish that this stochastic average is equivalent to the time ordered correlation function we obtain using canonical quantization, 6 V { m ) ) ~ fd[*)e-m ' ( L 9 ) To prove a general equivalence, Parisi and Wu must introduce more formalism from statistical mechanics. They point out that we may express the stochatic average of some function F[{z)] as; J X where P{4>(x);t] is the probability of having (x) at time t. Note that (1.10) just reexpresses definition (1.7) in terms of probabilities. If we now also recast the Langevin equation in a similar manner, we can show that the probability distribu-tion P(,t) evolves in time according to dnM tJd_\ 6* , s dt The above is known as the Fokker-Planck equation; the derivation for it can be found in standard texts. To simplify (1.11), we make use of an integrating factor and express the prob-ability distribution as P{,t] = e~25M$(#). The Fokker-Planck equation then reduces to: — = dt where „ 52 182S 1 /8S\2 . PART I: The Langevin Method of Stochastic Quantization 7 If S() increases fast enough as —• co, and if has countable number of degrees of freedom, then H has a discrete spectrum; n=0 where oo $ = ^ a » e " A n f ^ ^ ) - (1-13) n=0 We verify easily that $o = e - * 5 ^ is an eigenfunction of H with an eigenvalue of zero. This turns out to be the ground state since H is always positive. If the spectrum of H has a mass gap between the ground state and the next highest eigenvalue, we have in the limit t —• oo: which is the expression we obtain by canonical quantization. a) The free field two point correlation Now, let us return to the Langevin equation and consider a sample stochastic average. Take, for instance, {(pix^)^^,^)) for the free field (ie. the action is given by (1.1) with V(d>) = 0.) Imposing the boundary condition that at t = —oo, d> = 0 and 5(0) = 0, we have the solution to (1.5):^4) 4>(x,t) = J d[x, t)(y, t')} to order A 2 The crosses which appear in the diagrams are the points where the r/(x, t) 3 diagram" series. In Figure 1-1.1, we show the expansion to order A 3 . The bars on PART I: The Langevin Method of Stochastic Quantization 9 Figure 1-1.1 Tree diagram expansion to order A3. The bars on the branches indicate factors of r)(z,t). distributions join. A crossed line represents a factor of K(x,y); the normal two point propagator. An uncrossed line indicates the Green's function G(x — y) given by (1.15). The problem with this method of diagrammatic expansion, is that it generates many more diagrams than the canonical expansion. However, direct calculations, and certain combinatoric arguments lead me to suspect that all graphs of the same shape are equivalent, regardless of where the crosses appear. The proof of this will be a subject of future research. (1.3) Stochastic Quantization of Fermions: The Method of Fukai et al. Extending stochastic quantization to charged fields presents some additional complications; Parisi and Wu do not attempt it. To my knowledge, the first effort to stochastically quantize Fermion fields was that of Fukai et al. (5). t PART I: The Langevin Method of Stochastic Quantization 10 # +3 x ft + 2> Figure 1-1.2 Stochastic Diagrammatic Series to order A 2 The random sources join on the crossed lines. One source of difficulty is the anticommuting property of Fermi fields. There is no classical analog for anticommuting vector spinor and the classical concept of probability does not respect a Grassman algebra. Fukai et al. point out, however, that we can construct a consistent notion of probability which applies to Grassman numbers. They are able to use this definition as the foundation for a Fokker-Planck equation; which, we recall, was necessary to demonstrate an equivalence between stochastic averages and canonical correlation functions. A second problem is that if we directly apply the techniques we've used for scalar fields, we generate a Fokker-Planck Hamiltonian whose eigenvalues are not necessarily positive. This leads to stochastic averages which do not converge in the t —• oo limit. It is worthwhile to examine this second problem more closely. PART I: The Langevin Method of Stochastic Quantization 11 From the form of (1.5), we would expect the Langevin equations to be dtp dS . „ dip dS +, ^ where S = J ddxtf{ir) • d + im)rp. Let r),r}^ form a Markovian distribution, (i7(*,0> = («?t(*,0> = 0=(»7»7) = (»7V> {r){x, t)t]\x', t')) = 25{x - x')6{t - t'). Solving for V* and {ip^) we obtain; Jo e2t'(rP+im) _ j 7 • p + im Since 7 • p has both positive and negative eigenvalues, the expression for (ipip^) is not damped in the t —> 00 limit. Fukai et al. deal with this problem by starting out with a more general form of the Langevin equation and stochastic distribution; d^ dS , ^ a m = 1 d i A + r i M (r}(x,t)ri>(x',t')) = 2p-16(x-x')8(t-t'), (1.18) where we constrain a , / ? , 7 so {ipip^} converges in the limit t —• 0 0 . After some tedious calculation, Fukai et al. find that they can satisfy the constraint if they set a/?7 = 1. After more elaborate labor they define a Fokker-Planck Hamiltonian associated with the Langevin equation. Finally, they demonstrate that in the t —• 00 PART I: The Langevin Method of Stochastic Quantization 12 limit only the lowest eigenvalue contributes to the stochastic partition function. In this way,we recover the partition function for canonically quantized fields. While this method of quantizing fermions fields is mathematically sound, it cannot be combined with Parisi and Wu's theory to obtain a quantization technique for interacting fermions and bosons . Breit, Gupta and Zaks point this out in their paper "Stochastic Quantization and Regularization.''*6) The problem is that the fictitious time has dimensions of (£ ) 2 for the boson Langevin equation while it has dimensions of (£) for Fukai's fermion Langevin equation. (1.4) T h e M e t h o d of Breit et al. for Boson and Fermion Fields The solution forwarded by Breit et al. is to begin with a more general form of the Langevin equation; where K is a kernel independent of V ' J V ^ - One can show that the Fokker-Planck Hamiltonian associated with (1.19) is of the form; Breit et al. argue that as long as K is invertible and H is positive, the random average defined by (1.19) will have the canonical correlation function as its steady MMhW)) = 2K(x,y)6(t-t'), (1.19) where s t a t e d PART I: The Langevin. Method of Stochastic Quantization 13 The authors choose K = (17 • d + im) for fermions and K = 1 for bosons to obtain the Langevin equations; ^ = (32 + m 2)^ - iei •d^-ArP + Tj^ ^ = (3 2 + m V - i n • d(^l • Af + v ^ = 3 % - 5pt9 • A + etfirf + VA- (1-20) While the SQ procedure of Breit et al. appears to completely define Q E D , it suffers from some significant disadvantages. One shortcoming is the absence of a physical justification for the kernel K(x, y). The general definition of stochastic averages required to generate the (F\r)T)^]) correlations is t _ Jd[ri]F[rl\e-Jlivd+imr1i <*W1> = It is hard to conceive of this as a "stochastic distribution". It appears as though one could equally describe TJ and if as quantized Grassman operators which evolve according to the "action"; S = (^17 • d + tm)_1(|. Furthermore, recall that any invertible function that yields a positive Fokker-Planck Hamiltonian may be substituted as a kernel without disturbing the conver-gence of the theory. If we seek to explain SQ as anything more than a mathematical trick for mimicking correlation functions, the presence of a largely arbitrary kernel in our fundamental equation is a serious liability. Yet another unfortunate feature of equations (1.20), is that they leave the fictitious time with units of ( £ ) 2 while all other dimensions have units of (£) . The authors' original motivation for rejecting the SQ procedure of Fukai et al. was that the dimensions of the fictitious time should be the same whether we treat bosons or fermions. But rather than choose a quantization procedure which leaves one PART I: The Langevin Method of Stochastic Quantization 14 dimension fundamentally different from the others, it would seem more appropriate to design the Langevin equation so that all dimensions have units of length for both fermions and bosons. More specifically, we should be concerned about reworking Parisi and Wu's original formulation for boson fields so that it has the same order of derivative for all dimensions. Upon closer inspection, we find that this is not possible (at least so long as we are confined to only one fictitious dimension). If we try to choose the kernel in the generalized Langevin equation f)) = 2K(x, y)S(t-f) so that both sides include only first order derivatives it would have to be of the form * (*'y ) =(7V±im)g ( g-y )-But, this has a matrix structure not compatible with scalar fields. (1.5) S u m m a r y O u r discussion to this point has led us to a self consistent method of stochas-tically quantizing theories of interacting fermion and boson fields. Breit, G u p t a and Zaks use the bosonic Langevin equation of Parisi and W u together with their own fermionic Langevin equation. However, the method is not without problems of interpretation. We shall find in Part n that there is a simpler means of defining a stochastically quantized theory of interacting bosons and fermions. PART II: Toward a new Stochastic Formalism 15 P A R T II T o w a r d a new Stochast ic F o r m a l i s m To my mind, one of the most compelling features of the SQ technique is the way it hints at a physical basis for quantization . It intimates that a (d)-dimensional quantum theory is the large scale limit of a higher dimensional classical theory. On the surface, this is not much of a step forward. For, there is no a priori reason to suspect that classical theory is any more fundamental than quantum theory. Under the Copenhagen Interpretation, we accept the quantum behavior of both mechanics and field theory as a basic tenet of nature-classical theories appear as large scale approximations of an essentially quantum universe. An unfortunate consequence of this interpretation is that we can no longer derive all the dynamics of a system from its action: we must also postulate commu-tation and/or anticommutation relations. It would be more satisfying if we could devise a way of producing quantum theories without having to venture beyond the confines of a classical action. Here we find that there is an alternate, little used method of SQ which does just this for boson fields. We then go on to extend the technique to fermion fields. Finally, we introduce a possible theory of quantized interacting fields. In each case we embark from a classical action coupled to a random source and derive a quantum theory in lower dimension. PART II: Toward a new Stochastic Formalism 16 (2.1) A Field Equation for Bosons: The Method of Aharony et al. In 1976, A . Aharony, Y. Imry, and S. Ma* 1 0) demonstrated that a (d)-dimensional quantum scalar field theory is equivalent to a (d+2)-dimensional classi-cal field theory coupled to an external random source. They begin with the (d+2)-dimensional action (for convenience we label the two additional dimensions with the variable 'y'); S = J ddxd2y (k(x, y)4> + \(d»)2 + + V[fl) = J ddxd2yh(xiy) Now let (x, y) = -h(x, y). Now define a Green's function G(x, y) by, rf(x,y) = J dix'd2y'G{x-x\y-y')h{x',y') to obtain the momentum space Green's function equation; (k2+p2 + m2)G(p,k) = l, (2.2) where p is a four dimensional vector conjugate to x, and A: is a two dimensional vector conjugate to y. Solving (2.2) gives, G ( l - ^ y - y l ) = | - ^ _ ? ( t 2 + ^ + m 2 ) ( 2 . 3 ) Substituting this expression for the Green's function into the equation for (x, y) yields, V K 'y> J ; (27r)4(27r)2 (A: 2+p 2 + m2) ( M P , y W , y')) = ^ 4 ( P + P')62(y - y') With this we obtain, (<*(*, yM*',y')> = j d*xd2y j d^kd'pW eip(x-x)eip'(x'-x)eik{y-y)eik'(y'-y) * (fc 2+p 2 + m 2 ) 2 * PART II: Toward a new Stochastic Formalism 18 We now evaluate this expression at y = y' to get the expected result; (ttx,y)rt*,y)) = J ( 2 7 r ) 4 ( p 2 + m 2 ) When the theory is extended to interacting bosons, it generates a diagram-matic series which is pictorially identical to the series generated by the method of Parisi and Wu. (2.2) A Fermion Field Equation Our manifesto is to interpret a (d)-dimensional quantum theory as a limiting case of a higher dimensional classical theory coupled to an external random source. Let us begin with the case of free theory in four Euclidean dimensions. We describe the fermion fields by a classical action in five dimensions coupled to a random source S = j dhx (r)^rp + t)tf + ^(17 • d + tm)^) (2.4) Here,T7 and rfi are functions of x such that; Mzh V)) = 255(* - *') («?> = fa1) = 0 = {W) = fa V) where the above stochastic averages are defined as follows, Also, define 71,72,73,74 as the usual representation of the 4-D Euclidean Dirac matrices. We define 70 as the unit matrix. From the action (2.4), we derive field PART II: Toward a new Stochastic Formalism 19 equations (t'7 • d + im)rp = 17 ^f(t*7 • d + t'm) = »7+. (2.5) Upon inspection, we find that these field equations are equivalent to the Langevin equations proposed by Fukai et al. (with a particular choice of a,/?, 7). For example, consider (ip(x)ip^(x')) at XQ = xo' with the provision that rp{x) = 0 at XQ = — 0 0 . Solving, the appropriate Green's function equation, we have J (2ff) * ° + ^ ( * o ) , (2.6) where, rp(x,x0) = f dbyG^(x - y)r){y) tf{x,xQ) = J d 5yG^ T(x-y)i?(j/). Taking the stochastic average at XQ = xo' we obtain the usual expression for the time ordered propagator; (^ (x)^ (y)) = f £± J ( 2 7 T d 4p ijrl'-v) Furthermore, since the field equations we derived are the same as the Langevin equations posited by Fukai et al. , we may appeal to their work (using a Fokker-Planck Hamiltonian) to show that the stochastic averages converge properly to the canonical correlation functions when evaluated at xo = XQ'. PART II: Toward a new Stochastic Formalism 20 In this new formalism, much of the mystique surrounding the "fictitious time" disappears. The added dimension has the same footing in the action and field equations as the other dimensions.*9) We contract the additional dimension xo, not by taking a limit xo —* co, but by evaluating evaluating all field operators at the same point on the xo line. This method succeeds because XQ is now defined over the same range as the other dimensions. Can we repeat this trick and use a classical action to produce the fermion Langevin equations proposed by Breit et al. ? We would need to choose a classical action for fermions which has as its field equations (a2 + m2)ip + ^ The action we require is d S = r?^ + ijty + tf{d2 + m 2 - —U. ot Clearly, this is not a very palatable foundation Fermion field theory. Of the two methods we discussed in Part I for quantizing Fermion fields, that of Fukai et al. has the advantage that it is compatible with a simple higher dimensional action. While we have confined our discussion to the problem of generating a four dimensional quantum theory, the method is not so limited. In general, we may use a (d-f l)-dimensional classical action of the form (2.4) to produce a (d)-dimensional quantum theory for Fermions. dt at PART II: Toward a new Stochastic Formalism 21 (2.3) A Theory of Interacting Fermions and Bosons? We described how to stochastically quantize fermions and bosons respectively starting from a classical action of higher dimension. But if these results are to have any applications in Q E D , we must develop a theory which allows for fermion-boson interactions. An obvious obstacle to this, is the fact that we need two extra dimensions to quantize bosons but only one extra dimension to quantize fermions. I propose a theory in which (d)-dimensional fermions interact with (d—l)-dimensional bosons. I admit at the outset that I am not sure what physical applications such a theory could have. One possibility that springs to mind is e~,e+,7 interactions. Since the photon field is really confined to the surface of a three dimensional light-cone, we should be able to fully describe the behavior of photon fields with just three generalized coordinates. Only when the photon is absorbed must we expand our physics to include a fourth dimension. A counter argument is the success of Q E D , which describes photon fields in terms of four dimensions. On the other hand, gauge fixing really amounts to confining photon fields to three dimensional surfaces, or"orbits". For example, a common gauge is the Az = 0 gauge. At any rate, let us proceed. Consider the five dimensional classical Euclidean action; (12) (2.6) The field equations are — ( i i • d + im)rb + i e i • Arp —*pH*1' d + * m ) + »c^7 ' A iexp^ipip - 32Atl + d^d • A, (2.7) PART II: Toward a new Stochastic Formalism 22 where (v(xW(y)) = - y) = 25 5(* - y) {h(x)h(y))=26b(x-y). Solving for (rpip^) to zeroth order in e and evaluating at To = z'oj gives the usual four dimensional correlation function; (V(z)^(y)) = / ^ d4p t yp-(*-») r ) 4 7 • p — im' Solving for Af, to zeroth order in e, we obtain the Green's function W " J (2n)b * 2 where Avix) — j (PyGpvix - y)h„(y). Thus, we obtain; Unfortunately, I have not had time to proceed further with this theory. I hope to look at it in more detail in the future. PART III: Applications to Quantum Mechanics 23 P A R T I I I A p p l i c a t i o n s to Q u a n t u m M e c h a n i c s In Part II, we found that, by adding a random source to the action of a (d+2)-classical scalar field theory, we can generate the correlation functions of a (d)-dimensional quantum theory. It seems natural to wonder whether we may use the same technique to generate quantum mechanics from classical mechanics. Since the quantum scalar field may be thought of as a lattice of simple harmonic oscillators, we might expect that we can stochastically quantize each of the oscillators individually. So let us consider the harmonic oscillator in (l+2)-dimensions: where q — q(t,y), the mass of the particle is m and the frequency of oscillation is u>. Applying the Euler-Lagrange equations for q, renders the differential equation; (3.1) V2q(t,y) + u2q(t,y) = -$=h(t,y). (3.2) This can be solved easily, to give us, dt'd2y' d2k dp hit'.y'yv^eiHy-y') ( 2 T T ) 2 ( 2TT) {k2 + p2 + a,*) Now apply PART III: Applications to Quantum Mechanics- 24 {h(t, y)M*', y')) = 4**6{t - t')62(y - y') and obtain that at y = y', i (,\ r*'« 4 , r * /" * c p k e"""~''> W W ) ) - — J M l * ? + We are interested in the equal time correlation function, which is just the mean value of q2. So we set t = t', intergrate over k and find that { q } ~ mj (2ir)(p2 + u2) arctanj^ j T r m o ; Evaluate the above expression over the principal branch, and we get the ground state mean squared displacement; w ' 2mw On the other hand, integrate over n-branches, and we obtain < H) This agrees with the expectation value of mean squared displacement for a quantum oscillator in the n-th eigenstate; (n\Q2\n)=(n+^j mu) PART III: Applications to Quantum Mechanics 25 Let us now consider the correlation {p[t)p{t')), where p is the momentum conjugate to q. In this case, the result is less straight forward. First, we require an expression for p, the momentum conjugate to q, and note that it is defined by dL(t, y, q, q) = mq, (3.3) where L is the Lagrangian associated with the action (3.1). Hence, we have from (3.3) that p{t, y) = ^ j d t d y J — — 2 (JFC2 + P 2 + W 2 ) • We proceed as before and discover that for y = y'f dp Pv>c-f') (p(t)p(t')) = 4nmh J ( 2TT) {p2 + u 2Y The above integral is linearly divergent. However, when we appealed to the Fourier transform method to obtain a solution to the differential equation (3.2), we tacitly assumed that oscillating functions of the form e'px were damped at infinity. So, we make the identification etpx = lim £_ 0+ eStp~e^x. With this qualification, we derive that (p2) = rrihu) arctan(p)jo j mfiu which agrees with the expectation value of the mean squared momentum for a quantum oscillator in the n-th eigenstate. PART III: Applications to Quantum Mechanics 26 When we combine the results of our calculations for (q2) and (p2), we obtain the expected eigen-energies for the quantum oscillator. This is an encouraging result. Nonetheless, some important limitations of stochastic quantization persist. First, it is not clear why integrating over n-branches should give the n'th state correlation function. More importantly, our definition of (h) limits us to consider situations where the initial and final states are the same. Under the canonical scheme, we may also calculate the non-diagonal matrix elements; If the stochastic method is to generate these non-diagonal terms, it will be necces-sary to generalize the notion of a stochastic average. as a canonical partition function. If we then choose a finite time range of integration for x, only certain discrete momenta are allowed in correlation functon. For the case of the simple harmonic oscillator we would have; Most natural, would be to view J ddxd2y h(x,y)h(x,y) as a Hamiltonian and oo <<7(ri)<7(r2))= £ , « ^ ( n - r 2 ) where and where the range of integration is from 0 to T. PART III: Applications to Quantum Mechanics To obtain the eigenenergies, we need only apply the usual path integral meth-ods. To obtain the non-diagonal elements, we must confine the initial state to be different from the final state and apply the same methods. Conclusions We have followed two different quantization strategies and have found that each leads to its own theory of interacting fermion and boson fields. I have argued that the first of these, that forwarded by Breit, Gupta and Zaks, is contrived and not legitimately stochastic. While this method may prove convenient for some types of calculation, I cannot imagine that it has any deep physical significance. The bulk of its physics is locked up in an inscrutible kernel function which mysteriously drops out of a "random" distribution. The second method is, at this stage, more of a conjecture than a theory. But, if it should correctly describe Q E D , it would have some advantages over canonical quantization. For instance, all the physics of the theory originates in the action; there is no need to posit commutation relations. Also, in stochastic quantization, time ordering plays no role. I am skeptical of the role it plays in canonical quantization. Why should the temporal dimension enjoy such fundamental preeminence over the other dimensions? When we work in Euclidean space, all dimensions play equivalent roles in the action. So I find it surprising that time and inverse temperature should play such a distinctive role in defining correlation functions. Even if SQ can generate Quantum Electrodynamics, a further challenge re-mains. Can it can completely define quantum mechanics? To my mind, this is very much an open question. Nonetheless, it is clear from our discussion in Part III that a generalization of the notion stochastic averaging to finite domains must take place. PART III: Applications to Quantum Mechanics 28 Whatever benefits SQ has to offer, they come at a price; we must introduce a random source term in the action. Also, we must posit an 'unseen' dimension in terms of which our universe exists as a steady state. Perhaps beneath the realm of the quantum is a truly classical universe; a universe so buffeted by primal vaccuum fluctuations that it appears quantized in a long term or 'macroscopic' approximation. An ironic twist indeed! For, needless to say, a macroscopic approximation of the quantum universe takes us to a classical universe of one lower dimension than the one in which we started. Could it be that nature has carefully stacked the quantum universe within the classical, and the classical universe within the quantum, like a never ending series of Chinese boxes? 29 Footnotes (1) We may obtain expression (1.1) by interpreting the field as an infinite volume lattice of quantum harmonic oscillators. The coordinate l i s then just the index of the harmonic oscillator in the position (x\,X2,13). Also, (1.2) may be derived from the commutation relation for the quantum harmonic oscillator, [Xi,P}] — ihSij. (2) To avoid confusion, we will use small brackets for stochastic averages and large brackets for correlation functions. (3) The Langevin equation traditionally describes the time evolution of a system affected both by a random force and a frictional, or damping, force. It is the damping force which guarantees the approach to equilibrium. Later on, we will find it necessary to generalize equation (1.5). (4) Note that in the proof of equivalence using the Fokker-Planck Hamiltonian, we tacitly assumed an initial boundary of t = 0. A consequence of this assumption was that we only obtained an equivalence between stochastic averages and correlation functions in the limit t —• 00. If, on the other hand, we choose t — — 00 to be our initial boundary, the limit t —* 00 is irrelevant; as long as all field operators in the random average are evaluated at the same fictitious time, we obtain the expected correlation function. (6) J. D . Breit, S. Gupta, and A . Zaks, "Stochastic Quantization and Regularization Nuclear Physics B 2 3 3 (1984) 61-87. (7) Breit et a/.pp.62-63 (8) Ibid. p. 64 (9) It would be interesting to find out whether the physics of the theory is independent of which dimension we contract. I have not fully investigated this issue. (10) A . Aharony, Y . Imry and S . K . M a , Physical Review Letters 37 (1976), 1364. References 30 [I] G . Parisi and Wu Yongshi, Scientia Sinica, 24 (1981) 483. [2] T . Fukai, H . Nakazato, I. Ohba, K . Okano, and Y . Yamanka, Progress of Theoretical Physics, 69, 5, (1983), 1600. [3] J . D . Breit, S. G u p t a and A . Zaks, Nucl. Phys., B 2 3 3 (1984) 61. [4] A . Aharony, Y . Imry, and S. K . M a , Phys. Rev. Lett., 37 (1976), 1364. [5] A . Niemi and L . Wijewardhana, Ann. of Phys., 140 2 (1982), 247 [6] G . t'Hooft and M . Veltman, Nucl. Phys.,B44 (1972), 189. [7] A . Niemi and G . Semenoff, Phys. Rev. Lett. 51 (1983), 2077. [8] S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. 140 (1982), 372; A . N . Redlich, Phys. Rev. D 29 (1984), 2366. [9] S. Coleman and B . Hil l , Harvard University Preprint H U T P - 8 5 / A 0 4 7 [10] M . B . Gavela and H . Huffel, C E R N Preprint, CERN-TH.4276 /85 ; [II] H . Huffel and P . V . Landshoff, C E R N Preprint, C E R N - T H . 4 1 2 0 / 85. [12] R. Jackiw, Phys. Rev. D 29 (1984), 2375. APPENDIX : Stochastic Regularization and the Topological Mass in QED-8 31 A P P E N D I X Stochast ic R e g u l a r i z a t i o n a n d the T o p o l o g i c a l M a s s i n Q E D - 3 The problem of evaluating divergent Feynman Diagrams has led to a number of different regularization schemes; most notably the Pauli-Villar's technique and dimensional continuation. The vaccuum polarization graph in 2+1 dimensions is unusual in that both dimensional continuation and the Pauli-Villar's method fail to provide an unambiguous regularization of its parity odd component. Furthermore, the two regularizations give results which are incompatible with one another. This confusion suggests the need to examine results obtained by other techniques. Here we show that stochastic regularization provides a well defined method to evaluate the parity odd component of the vacuum graph. Consider the amputated Euclidean vacuum polarization graph in 3-D Q E D , n H x , y) = {tf(*b^(*)0(y)7^(y))eon.. where JI2 is the parity odd component and ^ ( 0 ) is the topological mass. Here, the 7^ comprise a 2 x 2 representation of the 3-D Dirac matrices; 7 0 = a 3 ; 7 1 = a 1 ; 7 2 = a 2 APPENDIX : Stochastic Regularization and the Topological Mass in QED-S where = 6pV — diag(l, 1, 1) and a3 are the Pauli matrices. In momentum space we have, to one loop, " " M " / frfo ^ ^ ^ r ( K -!>.) - im) ( 1 ) By power counting, this integral is linearly divergent. Pauli-Villar's regularization gives a topological mass, n2(0) = ^-(sign(m)-sign(M)) The presence of the sign(M) term indicates that the Pauli-Villar's fermions fail to decouple. Furthermore, the result is ambiguous since the sign of M is arbitrary. These difficulties cast doubt on the applicability of Pauli-Villar's regularization to this problem. On the other hand, regularization by dimensional continuation provides a topological mass n2(0) = ^-sign(m). Although this result is not in itself ambiguous, dimensional continuation is very suspect here. There is no unambiguous way to dimensionally continue T H ' ^ T * = 2\sfll/X. A similar problem appears in 4-D for T r / y ^ ^ y = 4 ie ' " / A p as has been discusssed by t'Hooft and Veltman'6'. There exist independent reasons for believing that the result obtained through dimensional regularization is correct. These have been discussed by Niemi and Semenoff'7!. Yet, there remains a need for a perturbative regularization which can unambiguously treat the parity odd component of the given diagram. Stochastic quantization as forwarded by Parisi and Wu t1' and stochastic reg-ularization, developed by Breit et. al., !31 offer a promising alternative. Stochastic APPENDIX : Stochastic Regularization and the Topological Mass in QED-S S3 quantization couples classical fields to markovian noise in an auxiliary time di-mension. Quantization appears as an equilibrium distribution in the large time limit. One regulates diagrams by substituting a non-markovian correlation which approaches the markovian case as a limit. We now apply this technique to the vacuum polariztion graph in 3 dimensions. We begin with the Langevin equations; tlx + (iD + im)r]2 (3) fji(iD + im) + fj2 (4) where D = 7'J(<9/i + A^) and r is the auxiliary time. Setting V = Ii + (iD + im)t]2 we attribute to 77 the correlation, (r?(x, t)fj(x', t')) = 2(iD + im)6d(x - x')a(t - t'). (5) Here, T) = 6 { x ~ y ) 6 { T ) which has a solution; G ( L , Y ' T ) = / ( ^ 8 E * ( * " " , " ^ + R A , ) ^ ( R ) - ( ?) Next, we substitute for rp and ip in the definition for Ti1"' and obtain, after some straightforward calculation, the regulated integral: n " " ( P ) = 4 ^ T dndT2dT3dTt J - ^ e ^ - ' l + r « ) ( ( P + * ) * + ™ » ) + ( 2 r - ^ + ^ ) ( t » + » » ) x Tr ( 7 " $ + im)7 I ' (^ - j6 + im)) a(r 2 - r3)cT(r4 - n). This regulated integral is finite, hence, we may take the trace to get n 2 ( p 2 , r ) = 8m fT dridndndu f - ^ e ^ - ' i + ' O l l p + t f + m ' ) Jo J l^ fl") x e ( 2 r - r 2 + r s ) ( f c 2 + m 2 ) a ( T 2 _ _ ^ Subsequently, noting that r has units of ^ 7 , we see that the integral is U . V . finite if we lift the regularization. Thus, we replace CT(T2 — T3) with 6(r2 — T3) to obtain; l k ( l _ e - ^ , W , ) ( l . e - 2 , ^ W ) ) n 2( P ,r) = 2 m y ^ ((p + fc)2 + m 2 ) ( A 2 + m J ) <"> APPENDIX : Stochastic Regularization and the Topological Mass in QED-S S5 Evaluation of this integral in the limit r —» oo yields, T T I 1\ ^ • P 2irp yjpi + 4 m 2 Furthermore, integrating equation (8) with p2 = 0, we evaluate the topological mass term for finite T ; n2(0, T) = —sign(m) sign(m) erfc(v/2m2T) - sign(m) erfcfv72m2r) 2m 2 N / 2 ^ I e - 2 r m 2 + - ^ ^ ^ ( m ) e rfc(v / 4m 2 T) m JT2 TT 1 T71 2 + —sign(m) erfcfv/imr) + -^VTTT e~ 4 m T . (9) 4TT Note that 17.2(0) = 0 at r = 0, and grows in stochastic time so that at at equilibrium sign(m) n2(o) = Air This agrees with the result predicted by Niemi and Semenofd7'. Also, this result for the topological mass 17.2(0) is exact; Coleman and Hill ' 9 ! have shown that there are no higher order corrections to the one-loop result. We expect that the parity-even component of the graph may be regulated in a manner analogous to that used by Gavela and Huffel' 1 0!. Thus, stochastic regularization provides a complete and unambiguous definition of massive 3-D Q E D . Also note that in a weak, smooth external field, the parity abnormal part of n,u/ yields the leading term in the induced current APPENDIX : Stochastic Regularization and the Topological Mass in QED-S 36 This grows from zero at T = 0 to its equilibrium value at T —> oo, j"(x) = ^-sign(m)e'"'AF I / A(x) + . . . Although stochastic regularization has proven valuable for for massive 3-D Q E D , there remains some ambiguity if we try to extend the above calculation to the massless limit. Equation (9) is a function of (m2T). The limits m —• 0 and r —• oo may be taken simultaneously with m 2 r = A, where A is an arbitrary positive constant. The topological mass then varies from 0 - » -^-sign(m) depending on the choice of A. The problem is further complicated by the fact that the limits p 2 —• 0 and m 2 —» 0 are not independent (as we have tacitly assumed.) The A dependence of the massless 17.2(0) indicates degeneracy of the quantum ground state as recognized by Jackiw [7]. The degeneracy will not be lifted by higher order corrections , since, as we have noted, the One loop topological mass is exact. The ambiguity may be relieved if we choose both U . V . and LR. regulators dependent on a stochastic time dimension. This remains a subject for further investigation.