l/ UCJ) 3ZU`—M"/Av (@3-*% S Lo ek A A UGDPHY/IIRPA 93-41 Wnm 1a1nii¤ifiiiE I PGGBELSB4 Inconsistency of Minimal Coupling in the Presence ofthe Square-Root Klein-Gordon Operator JoHN R,. SMITH Physics Department, University of California, Davis Davis, CA. 95616 [December 9, 1993] PACS (03.65.pm, 03.70.+k, 11.15.-1]} OCR Output WT! H. Weyl, Proceedings National Academy of Science 15, 323 (1929); H. Weyl, Physical Review 27, 699 (1950); H. Weyl, Space—Time—Matter, (Dover Publications, INC., New York, 1952), pp. 121-129 and pp. 282-295 (Note Weyl’s struggle to distinguish "geometrical” properties from "phase-fields.’ He obviously had some intuition that the non-Riemannian geometry he was proposing should not strictly be applied to vectors and tensors, but to phases. He was lacking the necessary insight from quantum mechanics to use imaginary phases at this time and apply the results to matter fields). R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity, (McGraw-Hill Book Company, New York, 1975), 2nd Ed., pp. 491-507; P.A.M. Dirac, Proc. Royal Society A133, 60; V. Fock Zeit. Physik 39, 226 (1927); F. London Zeit. Physik 42, 375 (1927); C. N. Yang, Annals of New York Acad. of Science, 294, 86-97 (1977). 10. Martin Lavelle amd David McMullan, Phys. Rev. Lett. 71, 3758 (1993). 11. John R. Smith, Second Quantization of the Square-Root Klein-Gordon Op erator, Microscopic Cansality, Propagators, and Interactions. UCD/HRPA 93-13. 10 OCR Output ABSTRACT lt is shown in this paper that the assumption of minimal coupling is incorrect given the representation of x/mz — V2 using Fourier transforms. In the literature cited in Refs. [1] and [5] minimal coupling, or a non—local generalization of minimal coupling, was the essential assumption used to prove violation of Lorentz invariance of x/m2 — V2 in the presence of interactions of the following form m2 + [—iV + eA(x)]2 Hence these proofs are inconclusive. 1. Minimal Coupling and the Exponential Shift The assumption of minimal coupling is that interactions can be represented by modifying the derivative operator as follows Gul/’ _* (gu 1 i€Au)d’> where Au is a four—vector potential representing the interactions and 1b represents the matter—field interacting with the four-vector potential. Au is further required to satisfy the local gauge transformation property W = €><i>{i€9(¢)}¤/n (2) A], : Al, — 3,,9. 1 show explicitly below that the above prescription breaks down in the general case of arbitrary functions 9(m) for the square—root Klein—Gordon operator. The only OCR Output rtw case where local gauge invariance works is when exp{ie0(x)} is an eigenfunction of V. Let h(k) : f[h(x)] represent the Fourier-transform operator _ _ —ih,(k) - F[h(x)]1 Qw)/f - 32 ek·:r h(x)d 3 m. (3) Then the Fourier transform of the product of two functions is represented by the convolution theorem r lf(¤?)9(¤5)l1 5gf(Z ——5f(k W) ·€)9(€)d`$·f (4) Using Fourier transforms the action of the square-root operator on 1/2 can be represented as follows mz · Y721/»(;c,t):1 ~—-(m/ MM/m2 + k2 i/i(k,¢)d—*k, i2 c > 5 1 · . I- —e"“( yly/m2 + k2 »¢(y,t)d3kd3y. »§ (2r) Consider multiplying 1/1 by a phaselthat is a function of space—time position where f (sc, t) : exp{ie0(x, By the convolution theorem of Fourier transforms the action of \/m2 — v2 on 1/¤’(:r, rf) reduces to <2¤>3Mm2 — V2 ¢’<x,¢> I / 6***%/mz + k2 / fo —<>¢<<,¢>d·sdk,’3 <¥“f<k> I 46%/m2 + Us + or ¢><s,¢>d<dk,33 (zw)3/2 / €*'*·xf(k)\/m2 + (12- my rp(x,r)d3k. (7) OCR Output ITi`ll ` The square—root operator therefore picks up a convolution over the wave—number in the expansion of the function We operate with a shifted square—root operator on ¢(a:,t) and integrate the shifted operator weighted by (i.e., the operator is shifted by the Fourier wave-numbers of f This is essentially an eigenvalue expansion of f over the complete set of plane—waves. Notice that the gradient of does not enter directly into the square-root operation. This is very similar to the exponential shift property of differential operators in the Heaviside operational calculus. In the Heaviside calculus a differential operator P(D), (where P(D) is a polynomial in the derivative operator) enjoys the following property known as the exponential shift €`"P(D)€'””f(¢¤) = P(D + r)f(w), (8) whereas the square—root operator obeys the following exponential shift law €”"°““” \/mz — V2 €"°`””¢(¤) = x/m2 + (hl — N)? M1) Given the above facts I will prove that the exponential shift property of the square-root operator is inconsistent with the transformation properties of the four potential as required by local gauge—invariance. In the case of local gauge transfor mations of the form where : exp{ie0(x)}, the gradient, V0(.2:), does not appear inside the square-root operator (except for the trivial case ee9(a:) = Since this is the essential assumption ofgauge—invariance for minimal coupling, minimal coupling will not hold true in the general case. The essential problem can be explicitly formulated by asking: Is minimal coupling consistent with gauge OCR Output WM invariance given the shift property in the convolution formula of Eq. (7)? In order for this to be the case insert the assumed form of minimal coupling in the square root operator and perform the convolution required to represent the operation of m2 — V2 on exp{éeH(x)}z,/J. l obtain the following condition for consistency —L— (2,,):s/2 e’k`zf(k)\/mz + (e1Zf'(:1:) + I; — iV)2 1/2(:c t) dgk * (10) exp{ie0(;v)}\/m2 + (eA(a:) — iV)2 1b(;z:,t). The transformed four—vector potential, AQ,(x,t), would therefore be required to transform as A(](x,t) = A0(:c,t) — 6"t9(:c,t), (U) ., . A (a:,t) = A(;z,t) — V9(a:,t) =s A(x,t) _— k/e, which requires that the function 9(;c,t) satisfy the relation 6V0 Z k. (12) This means that minimal coupling is not consistent with gauge invariance for the square—root Klein—G0rdon equation except for the case of specific transformation functions satisfying V exp{ie9(z)} = ikexp{ie9(a:)}. (13) The above inconsistency between the exponential shift property of the square—root operator, the Fourier convolution theorem, and the requirement of local gauge in variance is severe and renders applications of minimal coupling (e.g., violation of Lorentz invariance for the square—root operatorl in the presence of interactions) moot (except restricted by the above special case). Of course, the argument of this OCR Output IFN ‘ paper rests heavily on the assumption that the action of the square-root operator can be represented via Fourier transforms. Being that x/mz — V2 is a fractional op erator one must allow for the possibility that it is multi—valued (“half”—derivatives are also multi-valued and disagreements existed between different operator defini— tions for the value of the half—derivative of unity, i.e., lim(m_,0) and there— fore different representations of the operator acting on functions are possible. It is not beyond the realm of possibility that other representations of \/m2 — V2 would possibly allow minimal coupling. However, such definitions would fall outside of the realm of operators represented by Fourier transforms. The non—local representation of minimal coupling (see Ref. attempts to represent the gradient inside the square-root using the momentum—space represen— tation 1Y—>l2?—€A1(q»1¤)l. Where A,;(q,p) : / (d6;1c)]{(q,p,a:)Ac;(t,:r), Ag : V, (14) (ddr) K(q,p,:c) : 1. Exactly the same problem regarding gauge-invariance presents itself in the case of Eq. (14). The exponential shift property does not bring V9 inside the square-root and hence not inside the integral defining At(q,p). Therefore Eq. (14) does not admit gauge-invariance, since there is no way to transfer the addition of a gradient to AC;(t,;1:) from inside the square-root to a phase factor outside the square-root multiplying 1p(;r, t). The most interesting question is: ls there a generalization of the theory of interactions which will preserve gauge invariance in the presence of the squareOCR Output root operator? One could consider a change of variables in the arguments of the exponential of the Fourier transform that involve V0(;r), however in this case one leaves behind Fourier transforms and moves into the realm of Fourier Integral Operators] Many considerations of the square-root equation in the presence of interactions involving minimal coupling have been noted in the literature? Also a more gen eral form of minimal coupling has been considered`) and shown to be inconsistent with Lorentz invariance. Many articles are critical of the square—root equation because Lorentz invariance is lost in the presence of external fields assuming that minimal coupling is the correct way to introduce interactions. As I have shown above, minimal coupling in the context of the square—root Klein—Gordon operator is inconsistent with gauge—invariance. Hence the premise statedffl without proof in Ref. [1] is inconsistent. Therefore, the question of the Lorentz invariance of the square-root Klein Gor don operator in the presence of interactions remains an open problem. There may exist a representation that is both Lorentz-invariant and gauge—invariant. One such possible alternative representation of interactions was suggested to describe the Aharonov—Bohm effectu and developed extensively by S.