Proper-time Method Research Articles

New approach to quantum corrections in synchrotron radiation. II

Continuing our efforts to provide a simple and elegant derivation of a compact (exact) expression for the radiated power of the synchrotron radiation, we here generalize our previous discussion of the radiation by a spin-0 charged particle to that of a spin-\textonehalf{} particle. As in the spin-0 case, we compute the spectrum of the radiated power by using the proper-time method to first calculate the forward Compton scattering amplitude and then apply the optical theorem. The complication due to the inclusion of spin is handled by first decomposing the spin-\textonehalf{} states into the two subspaces of ${\ensuremath{\gamma}}^{0}$, and then using the connection between the two subspaces that is provided by the Dirac equation. Aside from the simple and straightforward operator procedures in computing the result, we also obtain a very simple and compact (one parameter) integral expression for the spectral and angular distribution of the radiated power. Its high-energy, weak-field limit agrees with that obtained from the conventional approach and that from the mass operator method.

Covariant “in-out” formalism for creation by external fields

A quantized scalar field under the influence of external scalar, electromagnetic, gravitational (or other gauge) fields is considered. Two Fock representations of the field algebra to describe scattering, binding, creation, annihilation etc. are covariantly distinguished with the help of a propagator. The latter is obtained by extending Schwinger's proper time method to the case of nonswitched external fields, with the boundary conditions specified by the introduction of a very natural Hilbert space.

Regularization, renormalization, and covariant geodesic point separation

One of the techniques used in quantum field theory in curved space-times to eliminate divergences in the vacuum expectation value of the stress tensor for quantum fields propagating on a classical gravitational background is called covariant geodesic point separation. Beginning with the Schwinger-DeWitt proper-time method we show how to discard divergences in the effective action by renormalization of the coupling constants in a classical gravitational action functional. We then demonstrate how to determine which terms in the vacuum expectation value of the stress tensor vanish when this renormalization is carried out. This is done using the point-separation approach. We give the form of these terms for spin 0, 1/2, and 1 fields, massive or massless, on an arbitrary curved background. The procedure used is covariant and introduces no ambiguities beyond those inherent in any renormalization scheme. We note the appearance of trace anomalies which arise due to the breaking of conformal invariance by the renormalization process and give the form of the anomalies for arbitrary space-time dimension.

New approach to quantum corrections in synchrotron radiation

Owing to a recent controversy over the importance of the second-order quantum corrections to the power spectrum of synchrotron radiation, we here reexamine this question by using a proper time method to calculate the forward Compton scattering of a real photon on a spin-0 charged particle in external magnetic fields. This approach, aside from being the simplest method found so far, has the feature that it resembles the classical procedure so that the quantum corrections can be identified at every stage of the computation. The option of being able to perform first the angular integration before doing the proper time integral enables us to simplify the calculation and to obtain the quantum corrections unambiguously. We find that the criterion for the quantum corrections to be important is the same as that found more than two decades ago.

Vacuum expectation value of the stress tensor in an arbitrary curved background: The covariant point-separation method

A method known as covariant geodesic point separation is developed to calculate the vacuum expectation value of the stress tensor for a massive scalar field in an arbitrary gravitational field. The vacuum expectation value will diverge because the stress-tensor operator is constructed from products of field operators evaluated at the same space-time point. To remedy this problem, one of the field operators is taken to a nearby point. The resultant vacuum expectation value is finite and may be expressed in terms of the Hadamard elementary function. This function is calculated using a curved-space generalization of Schwinger's proper-time method for calculating the Feynman Green's function. The expression for the Hadamard function is written in terms of the biscalar of geodetic interval which gives a measure of the square of the geodesic distance between the separated points. Next, using a covariant expansion in terms of the tangent to the geodesic, the stress tensor may be expanded in powers of the length of the geodesic. Covariant expressions for each divergent term and for certain terms in the finite portion of the vacuum expectation value of the stress tensor are found. The properties, uses, and limitations of the results are discussed.

Feynman quantization and the S matrix for spin particles in Riemannian spacetime

The Feynman space-time interpretation of quantum field theory is extended to the case of general Riemann space-time (external gravitational field). The expression in the form of a functional integral for the causal propagator is deduced by means of the proper time method and a correspondingly generalized path integral method. The interpretation of the Feynman diagrams in terms of particles and antiparticles is done in such a way that the usual space --time properties of the causal propagator (the forward movement in time of particles and backward movement in time of antiparticles) take place. The expression for the amplitude of the creation of the pair in the external gravitational field, including the instability and degeneration of the vacuum, is deduced. It is shown that vacuum instability is absent at least for closed 3-dimensional space. (auth)

Schwinger's Proper-Time Method and the Eikonal Approximation

Eikonal techniques are used to present an alternative treatment of the electron's Green's function in the presence of a plane-wave field. After studying the propagation of an electron in a prescribed electromagnetic field we eikonalize the corresponding Green's function. If the external field is chosen to be a laser field, the approximate eikonal propagator is turned into an exact one: Additional assumptions characterizing the eikonal approximation are automatically satisfied. The methods employed are Schwinger's pre-source-theory functional techniques.

The covariant Green functions of the photon and electron in weak gravitational fields

By adapting the proper-time method to both the electromagnetic and Dirac fields in interaction with a weak vierbein field, bein gauge is treated on an equal footing with coordinate gauge. The calculations of the local one-photon as well as the second- and first-order-differential one-electron Green functions are developed in terms of the first-order vierbein and auxiliary fields and the weak-field potentials, and are shown to exhibit manifest coordinate- and bein-gauge covariance. The gauge separations of the first-order fields allow the isolation and evaluation of the contributions from both the coordinate- and bein-gauge functions, permit expressions of the first-order contributions to the Green functions in terms of the gauge-invariant weak-field potentials, and lead to the modified versions of the Mandelstam-DeWitt path-dependent line integrals.

On covariant scalar green functions in a weak gravitational field

By applying the proper-time method to a scalar field in interaction with a weak gravitational field, the coordinate arbitrariness of the linearized gravitational theory is treated as the counterpart of electromagnetic gauge. The calculation of the scalar Green function is developed and shown to possess manifest coordinate-gauge covariance; the explicit contribution of the gauge function isolated by such a procedure is evaluated. A line-integral choice of form for the gauge function when re-expressed in terms of the fields produces the path-dependent Mandelstam line integral upon a change of base gauge. The space-like path averaging of gauge-independent line-integral expressions of the first-order fields does not correspond to a familiar gravitational gauge. Gauge-invariant gravitational potentials are constructed in the case of constant curvature. The exact expression obtained for the corresponding Green function exhibits the expected singularities of the flat space-time case.

Euclidean Quantum Field Theory. I. Equations for a Scalar Model

The analytic continuations to imaginary time of the Green's functions of local quantum field theory define Euclidean Green's functions. Use of the proper-time method allows to represent these functions as multiple Wiener integrals of functionals that obey infinite systems of coupled integral equations which are similar to, and for the particular model of a complex scalar field in quadrilinear self-interaction considered here a limiting case of, systems studied in quantum statistical mechanics by Ginibre. As a consequence, the Euclidean Green's functions can for this model be obtained by a limiting process, with temperature and density going to infinity, from the reduced density matrices of a nonrelativistic Bose gas. Reduced functionals are defined and their equations determined as a prepartory step to renormalization in the super-renormalizable cases of two and three dimensions.

On the Positron Theory of Vacuum

In our previous consideration, the so-called ambiguities of quantum field theory can be removed by the certain general conditions of integrals. These procedures are equivalent with the subtraction theory which subtract the ambiguous terms from the matrix element in question. In this paper, we again examine these subtractions from the slightly different view point and develop this consideration along the positron theoretical subtraction and indicate the gauge difficulties Can be removed by some technical procedure treating with the density matrix of the vacuum. To clarify it, we utilize the solution of density matrix in the constant external electromagnetic field obtained by Fock's proper time method and separate the non gauge. terms from it by this procedure and then arrive the throughout gauge invariant result. For the case of the density matrix in the arbitrary external field, we need the slight modifications, but there does not appear the essential changes . .of the subtraction terms.

On Gauge Invariance and Vacuum Polarization

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. Rigorous solutions of the equations of motion can be obtained for a constant field, and for a plane wave field. A renormalization of field strength and charge, applied to the modified lagrange function for constant fields, yields a finite, gauge invariant result which implies nonlinear properties for the electromagnetic field in the vacuum. The contribution of a zero spin charged field is also stated. After the same field strength renormalization, the modified physical quantities describing a plane wave in the vacuum reduce to just those of the maxwell field; there are no nonlinear phenomena for a single plane wave, of arbitrary strength and spectral composition. The results obtained for constant (that is, slowly varying fields), are then applied to treat the two-photon disintegration of a spin zero neutral meson arising from the polarization of the proton vacuum. We obtain approximate, gauge invariant expressions for the effective interaction between the meson and the electromagnetic field, in which the nuclear coupling may be scalar, pseudoscalar, or pseudovector in nature. The direct verification of equivalence between the pseudoscalar and pseudovector interactions only requires a proper statement of the limiting processes involved. For arbitrarily varying fields, perturbation methods can be applied to the equations of motion, as discussed in Appendix A, or one can employ an expansion in powers of the potential vector. The latter automatically yields gauge invariant results, provided only that the proper-time integration is reserved to the last. This indicates that the significant aspect of the proper-time method is its isolation of divergences in integrals with respect to the proper-time parameter, which is independent of the coordinate system and of the gauge. The connection between the proper-time method and the technique of "invariant regularization" is discussed. Incidentally, the probability of actual pair creation is obtained from the imaginary part of the electromagnetic field action integral. Finally, as an application of the Green's function for a constant field, we construct the mass operator of an electron in a weak, homogeneous external field, and derive the additional spin magnetic moment of $\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}}$ magnetons by means of a perturbation calculation in which proper-mass plays the customary role of energy.