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€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 ¢’ I / 6***%/mz + k2 / fo —<>¢<<,¢>d·sdk,’3

<¥“f I 46%/m2 + Us + or ¢>d

(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. Mandelstam] This representation of interactions can be traced back to earlier work by H. Weylv and the introduction of imaginary non—integrable phases? This representation uses non—integrable phases which have the property that mixed partial derivatives no

w longer commute. Recent workindicates that there exist non-local symmetries in

QED. Non—local symmetries could have significant implications regarding the the

#1 J. Sucher, op. cii., p. 22, Eq. (6.1). OCR Output ory of interactions. Therefore, it is most pertinent to investigate the consistency of

the Mandelstam representation as a non·local representation of interactions with

applications to general non-local Hamiltonians.

2. Discussion

Many years ago at the dawn of the formulation of relativistic quantum mechan

ics the square—root operator was considered in the construction of the relativistic

energy. Mathematical difficulties at the time resulted in abandoning the square

root operator equation and considering the operators as equations in their own

right and squaring them. It was also thought that relativistic invariance would

be impossible in the context of the interacting square-root operator because of

the apparent lack of symmetry between the time and spatial coordinates. These

issues are very important and since there is a mistake in the proof of violation of

u Lorentz invariance for the interacting square-root operator a re-investigationof

the general problem taking into account second quantization is important.

3. Conclusions

I have demonstrated the inconsistency of minimal coupling under the square root sign of the operator x/m2 — V2. This renders inconclusive all proofs of viola

tions of Lorentz invariance of the square—root Klein-Gordon equation that represent

interactions via minimal coupling.

Acknowledgements

This work was supported in part by the Intercampus Institute for Research at

Particle Accelerators (IIRPA) and USDOE grant DE FGO3 9lER40674. OCR Output

WU ” REFERENCES

J. Sucher, J. Math. Phys. 4, 17 (1963).

Kenneth S. Miller and Bertharn Ross, An Introduction to the Fractional

Calculus and Fractional Digerential Equations, (John Wiley and Son, Inc.,

New York, 1993), chapter 1.

Michael E. Taylor, Pseudodijferential Operators, (Princeton University Press,

Princeton, New Jersey, 1981), pp. 166-181.

see for example P.A.M. Dirac, Quantum Mechanics, (Oxford University

Press, New York, 1947), 3rd ed., p. 254; J. Sucher, op. cit.; Morse and

Feshbach, Methods of Mathematical Physics, (McGraw—Hill Book Company,

New York, 1953), Eq. 2.6.50, p. 256 (1953); J.D. Bjorken and S. D. Drell,

Relatiuistic Quantum Mechanics (McGraw—Hill Book Company, New York,

1964), p. 51; S.S. Schweber, An Introduction to Relatiuistic Quantum Field

Theory, (Row, Peterson and Company, Evanston, Illinois, 1961), p. 64.

H.J. Briegel, B.G. Englert, and G. Siissrnann, Zeit. Naturforsch. 46a, 933

(1991).

Y. Aharonov and D. Bohm, Physical Review 115, 485 (1959).

S. Mandelstam, Annals of Physics 19, 1-24 (1962); W. Drechsler and M.E.

Mayer, Fiber Bundle Techniques in Gauge Theories (Springer—Verlag, Berlin, 1977), p. 158.; Kerson Huang, Quarlcs, Leptons and Gauge Fields, (World

Scientific Publishing Co., Singapore, 1982), p. 74.

IV "U `