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The models have translation, Lorentz and gauge invariance and reduce to the conventional local quantum electrodynamics under the appropriate limit conditions, both the equations of motion of the charged particle and electromagnetic field obtained by the action principle lead to the normal form of current conservation.
Quantization of the models is realized by taking advantage of the formalism based on the Yang-Feldman equations and the Lehmann-Symanzik-Zimmermann reduction formulas. Finally, we employ a special choice of the models to calculate the vacuum polarization as an example to demonstrate the possibility of establishing a theory of quantum electrodynamics without divergence. There is a large literature on nonlocal field theory for a review and references, see refs 1 , 2.
One of the motivations for investigating nonlocal field theory is to establish a theory without divergence; since the renormalized theory of local gauge fields leads to finite predictions and to an unprecedented agreement between theory and experiment, if divergences still appear in the corresponding nonlocal theory and must still be eliminated by a renormalization procedure, then such a nonlocal field theory is not interesting. Of course, according to the contemporary point of view, any field theory must be renormalized; if a nonlocal field theory involves finite renormalization constants, then such a theory accords with the contemporary point of view and standard mathematical theory at the same time, since standard mathematical theory allows ignoring small or finite quantities but ignoring an infinitely large quantity does not accord with standard mathematical theory 3.
Some models of quantum electrodynamics QED with nonlocal interaction have been proposed 4 , 5 , 6. In this paper, we present a class of models of QED with nonlocal interaction. After looking back on some known nonlocal models of QED, we present the action of a class of models and the equations of motion of a charged particle and electromagnetic field obtained by the action principle.
Translation, Lorentz and gauge invariance of the models are proved, and we prove that both the equations of motion of a charged particle and electromagnetic field lead to the normal form of current conservation. The models are reduced to the conventional local QED under the appropriate limit condition.
Similar to some known nonlocal modes of QED, for guaranteeing the gauge invariance of the theory, the form of the models presented in this paper is far from that of conventional field theory. Next, based on the fact that a free charged particle and free electromagnetic field still obey the local Dirac equation and the local Maxwell equation of free fields, respectively, quantization of the model is realized by taking advantage of the formalism based on the Yang-Feldman equations and the Lehmann-Symanzik-Zimmermann reduction formulas.
Some properties, e. The models presented in this paper provide a wide range of choices; we employ a special choice of the models to calculate the vacuum polarization as an example to show the possibility of establishing a theory of QED without divergence. All symbols and conventions of this paper follow ref. The theory is invariant under the gauge transformation. Thus, H. McManus in ref. Specifically, by choosing , where r 0 is a constant, a theory of classical electrodynamics without singularities is obtained in ref.
However, M. Chretien and R. Peierls in ref. On the other hand, , in which S e and are given by 2 and 3 , respectively, can be written in the form 8. Although the form of this theory is far from that of conventional field theory, the theory is still invariant under the gauge transformation 4 and 5. For this theory, current conservation is no longer the normal form 6 , and there still exist divergences in the theory 5. Scharnhorst in ref.
Hence, the corresponding inverse matrices and exist and satisfy. For this choice, 16 becomes. It is obvious that 28 is invariant under the transformation given in 4 and It is easy to prove that the equation of motion 29 of a charged particle leads to the current conservation 6 ; on the other hand, as is well known, a very important and basic property of the equation of motion of the electromagnetic field of the conventional local QED is that which leads to the current conservation 6 by the approach.
We prove that 28 has the same property. Thus 31 becomes. From the above result, we obtain the current conservation 6.
From 28 and 29 , or 32 and 33 , we see that the free charged particle and free electromagnetic field still obey the local Dirac equation and the local Maxwell equation of free fields, respectively.
For this theory, so far, the unique method of quantization is to employ the formalism based on the Yang-Feldman equations 11 , Here, we list only the main steps of quantization of the models as follows.
In the last step, we include brief discussions, since we shall employ the Lehmann-Symanzik-Zimmermann reduction formulas instead of the S -matrix. And, furthermore, considering the asymptotic conditions 35 , from the equations of motion 30 and 28 we obtain the Yang-Feldman equations 7 :.
The last step of the method of quantization employing the formalism based on the Yang-Feldman equations is to introduce the S -matrix by. Conversely, for the theory with nonlocal interaction, we can prove that the LSZ reduction formulas 7 , 13 , 14 still hold. Hence, we employ the LSZ reduction formulas instead of the S -matrix to calculate the transition amplitude.
For example, for the case that the initial and final states are an electron, the corresponding transition amplitude is. A question naturally arises from the above discussions: now that the transition amplitude is only an element of the S -matrix, if the S -matrix defined by 39 does not exist, or exists but is not unitary, then is the transition amplitude determined by the LSZ reduction formula correct?
Here, we only address this question briefly. Notice that there are in fact two types of S -matrices: one is defined based on operators, e. The differences between the two types of S -matrices are subtle.
For example, the asymptotic conditions 35 based on operators are in fact incorrect; the correct forms of the asymptotic conditions are based on the state vector 7 , 13 , 14 :.
The exact expressions of the asymptotic conditions can be found in ref. Hence, regardless of whether the S -matrix based on operators exists, we can introduce the type of S -matrix based on the state vector, and, in addition, take advantage of the LSZ reduction formula to determine the element of this type of S -matrix, i. We therefore obtain a complete quantum theory of the models of QED with nonlocal interaction, as presented in the above section.
However, although we employ the LSZ reduction formulas instead of the S -matrix, basic principles such as Lorentz and gauge invariance, unitarity and causality of the S -matrix must be satisfied, of course.
We now ask that the elements of S -matrix, i. And, furthermore, according to the LSZ reduction formulas, the question of whether the elements of the S -matrix satisfy these principles is reduced to investigating the properties of such Green functions as. For example, it is obvious that if such Green functions as are Lorentz and gauge invariant, then the elements of the S -matrix are as well.
And, the condition of unitarity corresponding to that of the S -matrix is now , which also leads to the investigation of the properties of such Green functions as. As for the condition of causality of the S -matrix, following N. Bogoliubov and D. Shirkov 17 , we write the interaction action 16 in the form. Therefore, determining whether the elements of the S -matrix satisfy the condition of causality. On the other hand, experiences with some known theories show that the properties mentioned above of Green functions can be investigated only after the theories are researched deeply.
For example, if we employ the Coulomb gauge in the conventional local QED, then Lorentz and gauge invariance of the elements of the S -matrix can be proved only after the rules of the Feynman diagrams of the theory are obtained see ref. A different example is unitarity of the S -matrix of the theory of the non-Abelian gauge field: although as early as , R.
Feynman pointed out that one must add some additive terms in the theory of the non-Abelian gauge field to guarantee unitarity of the S -matrix under Feynman gauge 18 , and even though a general approach of adding additive terms was given by L. Fadeev and V. Popov 19 , the unitarity of the S -matrix of the theory of the non-Abelian gauge field can be proved only after Slavnov-Taylor identities are established.
What should we do when the models presented in this paper do not satisfy unitarity or and causality? A revelation from the theory of the non-Abelian gauge field is that, if the elements of the S -matrix of the models presented in this paper do not satisfy unitarity or and causality, then maybe we can add some appropriate additive terms to restore unitarity and causality, similarly to the addition of ghost fields to restore the unitarity of the S -matrix in the theory of the non-Abelian gauge field.
These questions will be studied further. Concrete calculation of the transition amplitude based on the LSZ reduction formula is lengthy and complex even for the conventional local QED. On the other hand, G. The concrete calculation process is lengthy and complex; here we only present the main steps and results.
We can prove that the first term in 48 is exactly the same as the results 2. Schwinger in ref. The method we use is to calculate straightforwardly. This method can not only prove J. The calculation result of given by J. We follow the above approach to deal with given by 50 ; for the case that no real pair creation has occurred, we finally obtain. By choosing constants and satisfying. The third term new divergent term in 55 arising from the new theory can be eliminated by imposing Lorenz gauge on external electromagnetic field.
It is possible that some terms arising from are similar to the first three terms in 55 , and, thus, contribute to the first three terms in Generally speaking, since the new theory brings many new terms, it can be predicted that new divergent terms in the new theory presented in this paper will appear. Just as the method of higher covariant derivative regularization of gauge theories 23 , 24 , 25 removes divergences in the original theory while at the same time bringing many new divergent terms, of course, all divergences are removed finally.
We predict that the divergences brought by new terms arising from the new equations of motion 28 and 30 can be removed finally. As for the calculation approach from 43 to 48 , which not only leads to a complex computation process, but also causes some divergent integrals, e.
Can we find a calculation approach such that there is no divergence in the calculation result? This question will be studied further. Both 21 and 23 play important roles in the proofs of the gauge invariance of the model and the conclusion that the equation of motion of the electromagnetic field 28 leads to the current conservation 6. Eqs 21 and 23 can be proved by a basic function algorithm.
Here we prove them by straightforward calculation. Some expressions obtained in the proof process are also used in the calculation of the vacuum polarization. According to 13 ,. By a straight forward calculation, we obtain the determinant V y and the inverse matrix of as follows. Therefore, , and according to the above expressions of , we can verify that.
Authors: Schwinger, J. Publication Date: Fri Jun 01 EDT Research Org.: Originating Research Org. not identified. OSTI Identifier: .
This paper is based on the elementary remark that the extraction of gauge invariant results from a formally gauge invariant theory is ensured if one employs methods of solution that involve only gauge covariant quantities. We illustrate this statement in connection with the problem of vacuum polarization by a prescribed electromagnetic field. The vacuum current of a charged Dirac field, which can be expressed in terms of the Green's function of that field, implies an addition to the action integral of the electromagnetic field. Now these quantities can be related to the dynamical properties of a "particle" with space-time coordinates that depend upon a proper-time parameter. The proper-time equations of motion involve only electromagnetic field strengths, and provide a suitable gauge invariant basis for treating problems.
Based on order one-loop effective Lagrangian derived from the 2-point photon vertex in quantum electrodynamics, we obtain a quantum modified Maxwell equations, and the classical expression of retarded potential is consequently modified by these equations. The results indicate that, due to the time-space non-locality of vacuum polarization, the vacuum polarization current is delayed relative to the field variation and induces a series of additional retarded potentials except for the classical part of retarded potential. Particularly, compared to the classical potential, these additional potentials are further retarded.
One of the biggest problems in cosmology today lies in the discrepancy between the high-z calculated and measured value of the Hubble constant H 0. Slides PDF. The Maldacena conjecture suggests that superconformal symmetry plays an important role in holographic theories.
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Abstract We improve on a method to compute the fermion contribution to the vacuum polarization energy of string-like configurations in a non-Abelian gauge theory. We establish the new method by numerically verifying the invariance under a subset of local gauge transformations. This also provides further support for the use of spectral methods to compute vacuum polarization energies in general. Numerical results for the physical on-shell scheme are also presented. We improve on a method to compute the fermion contribution to the vacuum polarization energy of string-like configurations in a non-Abelian gauge theory. We confirm that the vacuum energy in the MS renormalization scheme is tiny as compared to the mass of the fluctuating fermion field.
The course also covers the concepts of topological insulators and superconductors that have become an important part of the vocabulary of modern condensed matter physics. The goals of this review are to briefly introduce the physics of topological insulators to a chemical audience. Here we predict the first material realization. Condensed matter physics theory. A Brief Guide for Beginners. Imported questions retain onl.
We illustrate this statement in connection with the problem of vacuum polarization by a prescribed electromagnetic 6eld. The vacuum current of a charged Dirac.
Don't have an account? This chapter chronicles Schwinger's research in relation to Green's function, his first trip to Europe, and his work on the gauge invariance and vacuum polarization, the quantum action principle, electrodynamic displacements of energy levels, quantum field theory, and condensed matter physics. Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service.
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