Feynman Rules and Factor Ordering in Derivative ...

1693 _ Progress of Theoretical Physics, Vol. 48, No. 5, November 1972

Feynman Rules and Factor Ordering in Derivativ e Coupling and Spin-One Theoriest l Joel S. KVITKY*l and John 0. MOUTON**>

Departmen t of Physics, University of California Los Angeles, California 90024 (Received April 28, 1972)

§ l. Introductio n Recently there has been a revival of interest in the rigorous derivation of Feynman rules for canon,icaL theories of massive vector particles and theories with interaction terms q.uadratic in velocities. In an earlier paper treating the electrodynamics of charged vector mesons with arbitrary magnetic moment, Lee and Y ang 1> obtained the surprising result that the normal-depe ndent terms arising in the interaction Hamiltonian and in the propagators do not mutually cancel; a similar result obtains for chiral theories of pions when expressed in general gauge group coordinates. 2 >-•> In such cases, unitarity and Lorentz covariance are not manifest, and great care is required in developing the perturbation series to determine whether in fact these properties are preserved. We shall outline m

§ 2 the elements of the Lee-Yang theorem.

In was shown in a previous paper"> that. the application of tpe results of Ref. 1) to the massive Yang-Mills theory with arbitrary gauge group yields manifestly covariant Feynman rules; this result was then extended6l to the chiral massive gauge theory as well. It was indicated in Ref. 5) that the results of Lee and Supported in part by the National Science Foundation. Present address : R and D Associates, Santa Monica, California 90403. **l Howard Hughes Doctoral Fellow. Present address: Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024; and Hughes Aircraft Company, Canoga Park, California 91304. Tl

*l

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Several methods may be employed to derive the Feynman rules for canonical theories. With emphasis on vector particle theories, we recast the theorem of Lee and Yang into functional integral and Hori operator formulations. For the class of Lagrangians considered, equivalent results are obtained: The canonic~! generating functional is used to relate T and T* products to the Hamiltoniim and- Lagrangian versions of theories with dependent fields. With- the aid of Hori operators, we show that there is a wide class of ordering prescriptions compatible with the_ Lee-Yang. theorem. .Of these, the. only familiar candidates are .normal ordering and symmetrizatio n of momenta and .coordinates. It is argued that the latter is preferred, thus confirming a result of Suzuki and Hattori. Finally, the expanded Lee-Yang theorem for general ordering prescriptions is outlined, and ordering for a Yang-Mills theory is discussed.

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J. S. Kvit ky and J. 0. Mou ton

Yang also follow from the canonical pathinteg ral meth od when the integ ratio n over mom enta is carri ed out. In that pape r a nonr elativ istic theor y was cons idere d whic h exhib its the gene ral struc ture of the .field theor ies of inter est. In § 3, we shall discu ss in detai l how this prog ram proce eds for vecto r parti cle theor ies, whic h are subst antia lly comp licate d by the prese nce of depe nden t fields. We shall demonstrat e direc tly from the cano nical gene ratin g funct ional how the trans ition from T to T* prod uct is relat ed to the trans ition from Ham ilton ian to Lagr angia n form ulatio ns.

§ 2. The Lee-Yang theo rem In cano nical mass ive vecto r parti cle theor ies and theor ies wher ein the inter actio n Lagr angia n conta ins term s bilin ear in the time deriv ative s of the fields, one gene rally enco unter s normal~dependent term s in the inter actio n Ham ilton ian and· nonc ovari ant prop agato rs. In parti cular , if ¢,.-d enote s fJ,.¢ (r/J, a scala r field of mass m) or MV,. (V,., a vecto r field of mass M>O ), one has

(OIT [0,.( .x)0. (0)] IO)= D,..( .x) -ihg,.og.dJ4(.x), D,..( .x)=( OIT* [0,.( .x)0. (0)] IO), = -fJ,.fJ.JF(.x, m')

if 0,.=fJ,.¢,

(2·la ) (2·lb ) (2·1c )

= (M'g,..-fJ,.fJ.)JF(.x, M')

if 0,.=M V,., (2·1d ) wher e JF(.x, m 1) is the prop agato r ("co ntrac tion" ) for a mass ive scala r field. In orde r to treat the elect rody nami cs of charg ed vecto r parti cles in whic h the above situa tion arise s, Lee and Yang prov ed the follo wing theor em:· Cons ider a phys ical syste m

(2·2)

wher e (/) is a colum n vecto r made up of f local Herm itian fields

(2·3)


Unti l recen tly, the ques tion of facto r orde ring in these vario us theor ies had not been raise d. Som e start ling cons eque nces of these cons idera tions have been point ed out, with emph asis on chira l pion theor ies, by Dow ker and May es 7> in the conte xt of path integ rals and by Suzu ki and Hatt ori 8> who use the Feyn manDyso n-Wi ck expr essio n for the S matr ix. In §§ 4 and 5, we shall show how the Lee-Yang theor y arise s in a form ulati on based on Hori oper ators ; the most reasonable orde ring presc riptio n cons isten t with the earli er pape rs will be identified as total symm etriz ation of mom enta and coord inate s, in subst antia l agree ment with Ref. 8). Meth ods for impl emen ting diffe rent orde ring presc riptio ns will be detai led and orde ring for Yang -Mill s fields will be discussed.

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and A= A, B, and C are (fxf), (fx 1), and (1 X 1) matrix quantities which depend on the remaining fields in the theory. Let D(x) be the matrix contraction G0 Va),

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(3·1la) (3·1lb) (3 ·llc)

ITN(x) =, IIexp( -i~H/h) 3!

= exp

3!

[-i_h fd 4x~HJ.

(3 ·12)

Since iJH is 'constructed from the interaction part of the Lagrangian density, it will contribute to the perturbation expansion of the LGF unless it vanishes. The perturbation series arises from (3 ·lla) by expanding the exponential containing the part of the action which depends on the coupling constant; this will include (3·12) when iJH=/=0. A similar thing is done for the CGF in (3·6a) where the action is exactly as written in (3·6b). Following the discussion of Appendix B, "contractions" will appear in these two expansions which we wish to identify as T* and T products respectively. One can conClude from (3 ·lla) that

iJ ~ Z (1)/ ho' (4·1)

Here Hz is the interaction Hamiltonian in the interaction picture, N represents normal ordering and T~~. is a functional differential operator of the general form (4·2a)


for the operators in Schroedinger's equation. In fact, it has been shown7>' 10l that for certain physical systems for which a given ordering is preferred, any estimate for iJS may require the addition of a compensating term to the Lagrangian or Hamiltonian in the functional integral so that the correct form of the Schroedinger equation results. This compensating term is typically proportional to h 2 and as such resembles (and, in fact, may actually be) a commutator term, that is, a term which arises in the quantum Hamiltonian or Lagrangian as a result of rearranging factors via the canonical commutation relations. The precise form of this term is by no means unique, as one can easily understand: Suppose an estimate for iJS, arrived at by some specific rule for estimating the short-term action, leads to a particular ordering in Schroedinger's equation. One may achieve a modified ordering by one of two methods. One may search for a different rule for estimating iJS; conceivably, however, no appropriate rule may exist for the ordering desired. Alternatively, the same rule may be retained but one must add to H or L the classical transcription of the commutator terms relating the desired ordering to the original prescription. For the physical systems we consider, these commutator terms are never quadratic in momenta or velocities; it is for this reason that the form of iJH is independent of ordering questions. However, because of the ambiguity in the explicit form of the action, one is at a loss to describe precisely which terms in the various forms of the action within the functional integral actually participate in the perturbation expansion of the theory when expressed in Feynman diagrams. We have therefore chosen to present the factor ordering problem in the context of a formulation more reminiscent of familiar field theoretic methods. With the developments in the subsequent sections, it will be seen that the matter of ordering, particularly that implicit in the work of Lee and Yang, can be clarified.

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J. S. Kvitky and J. 0. Mouton

(4·2b)

.

.

(4 ·3a)

H=Ho+Hz =HPP+QQ ) +H(P-B) (I+A)- 1 (P-B) -PP-2C], (4·3b) where A= A, B and Care (fxf), (fx 1), and (1 X 1) matrices which depend on the coordinates Q alone. The contractions appearing in Eq. ( 4 · 2b) are: Ll!,,. (t) = ha2 and h~, respectively. This amounts to a rather minimal modification of the contractions, which are defined only up to addition of quasilocal operators such as lJ(t) or (d/dt)l)(t). 17 l One might conclude at first guess that the S matrix is independent of a~> a 2 and a 8 ; of course, this is not so. For ordering purposes, itis apparent that the operator T,. is changed by a multiplicative factor given by

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Acknowled gement We wish to express our appreciation to Professor R. ]. Finkelstein for his encouragem ent and guidance during the course of this work.


which is not based simply on factor ordering of P's and Q's.) Since symmetrization is associated with the equal-time contractions that one would normally accept, it can be said that NT,.[···] is identified preferably as symmetric ordering, unless some hard-boiled partisan insists on normal ordering. With this reservation, our result agrees with the claim of Suzuki and Hattori. How, then, does one implement alternative, nonexotic ordering rules in the derivation of the preceeding section? One can accomplish this by adding to H-the classical Hamiltonia n-the classical transcriptio n of the commutator terms relating symmetric ordering to the desired prescription ; one employs the equal-time contractions as they appear in (5 · 2a, b, c). By treating these commutator terms as part of the Hamiltonian , the derivation of the preceeding section remains valid and the form of (JH is unchanged. This is the analog of the Dyson-Wick procedure developed in the paper by Suzuki and Hattori. The chiral pion theory treated in that paper serves as an admirable example, and the reader should examine it for illustration of this point. For that field-theoret ic case, it is found that an ordering prescription distinct from symmetriza tion is required to ensure invariance under change of coordinates in the gauge group manifold. The relevant commutator terms, which are proportiona l to [h(J 3 (0) ] 2, are necessary elements in the perturbation expansion. Let us now touch briefly on the subject of quantum ordering in a Yang-Mills theory. For a general approach to this topic, one should refer to the paper by Fickler and Russo/ 8> where they derive the complete quantum action principle in their treatment. For our purposes, we shall be content to merely specify the ordering which leads to the earlier results for the massive Yang-Mills theory. Note the form of the Lagrangian in equations (3 ·la, b). Due to the total antisymmetry of the structure constants fabc' we find that the field intensities Va"~(x) are free of ordering ambiguity. Since these quantities enter in an intrinsically symmetric way in the Lagrangian in (3 ·la), it is evident that a quantum transcription of this Lagrangian is properly ordered for purposes of applying the Lee-Yang theorem; that is, it is symmetric in "velocities" and "coordinate s". (Remember that the time-compo nents of the fields and field intensities act as velocities in this theory;) The seemingly peculiar results found in Ref. 8) for the chiral pion theory do not arise for a Yang-Mills type of gauge theory. For the Yang-Mills theory, it is known 5> that the coefficients of the bilinear products of "velocities" are at most quadratic in the remaining fields. Therefore, the commutator terms relating the admissible Hermitian orderings of the Yang-Mills theory will at worst be (divergent) constants/9> which may be ignored.

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18) 19)


8) 9) 10) 11) 12) 13) 14) 15) 16) 17)

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