Relativistic field theory

Pauli W. Relativistic Field Theories of Elementary Particles. Rev. Mod. Phys. 13,... 

Following the common approach in relativistic field theory, which aims at a manifestly covariant representation of the dynamics inherent in the field operators, so far all quantities have been introduced in the Heisenberg picture. To develop the framework of relativistic DFT, however, it is common practice to transform to the Schrodinger picture, so that the relativistic theory can be formulated in close analogy to its nonrelativistic limit. As usual we choose the two pictures to coincide at = 0. Once the field operators in the Schroodinger-picture have been identified via j/5 (x) = tj/(x, = 0), etc, the Hamiltonians He,s, Hy s and are immediately obtained in terms of the Schrodinger-picture field operators. 

Engel, E. Muller, H. Speicher, C. Dreizler, R. M. Density functional aspects of relativistic field theories. In Density Functional Theory , vol. 337 Eds. Gross, E. K. U, Dreizler, R. M. Plenum Press New York, 1995, 65-118. 

Liiders, G. (1954) On the equivalence of invariance under time reversal and under particle-antiparticle conjugation for relativistic field theories, Dan. Mat. Fys. Medd., 1954 (28) 1-17. 

General principles of relativistic field theory require invariance under the combined transformation CPT. The simplest tests of CPT invariance are the equality of the masses and lifetimes of a particle and its antiparticle. The best test comes from the limit on the mass difference between and K. Any such difference contributes to the CP-violating parameter e. Assuming CPT invariance, 0g, the phase of e should be very close to 44°. (See the Note on CP Violation in Decay in the Particle Listings.) In contrast,... 

A. R. Bishop, D. K. Campbell, K. Fesser, Polyacetylene and relativistic field-theory models, Molecular Crystals and Liquid Crystals 1981, 77, 253. 

Within the Born-Oppenheimer approximation, we still need to know that the nuclear position parameters really correspond to the distances and angles of a classical molecular framework. Our choice of the Coulomb gauge ensures this—the nuclear positions only appear in the electron-nucleus interaction terms, and the derivation of this potential from relativistic field theory shows us that it is indeed the quantities of normal 3-space that appear here. Thus, any potential surface that we might calculate on the basis of the Born-Oppenheimer-separated electronic molecular Dirac equation is indeed spanned by the variations of molecular structural parameters in the usual meaning. 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

I restrict my attention to non-relativistic pioneer quantum mechanics of 1925-6, and even further to the time independent formulation. Numerous other developments have taken place in quantum theory, such as Dirac s relativistic treatment of the hydrogen atom (Dirac [1928]) and various modern quantum field theories have been constructed (Redhead [1986]). Also, much work has been done in the philosophy of quantum theory such as the question of E.P.R. correlations (Bell [1966]). However, it seems fair to say that no fundamental change has occurred in quantum mechanics since the pioneer version was established. The version of quantum mechanics used on a day-to-day basis by most chemists and physicists remains as the 1925-6 version (Heisenberg [1925], Schrodinger [1926]). 

S. S. Schweber, Introduction to Relativistic Quantum Field Theory, Section 6, Harper and Row, New York, 1961. 

In non-relativistic Schrodinger theory every component of the orbital angular momentum L = r x p, as well as L2, commutes with the Hamiltonian H = p2/2m + V of a spinless particle in a central field. As a result, simultaneous eigenstates of the operators H, L2 and Lz exist in Schrodinger theory, with respective eigenvalues of E, l(l + l)h2 and mh. In Dirac s theory, however, neither the components of L, nor L2, commute with the Hamiltonian 10. 

We teach our students that the infinite zero-point energy that arises when a free relativistic scalar (or Dirac-) field theory is canonically quantized can be subtracted (discarded) by a suitable redefinition of the energy-origin in other words, by normal ordering. However, the energy-origin can be re-defined only once, and only in homogenous space (i.e. without boundary conditions) and... 

Abstract The equation of state (EOS) of nuclear matter at finite temperature and density with various proton fractions is considered, in particular the region of medium excitation energy given by the temperature range T < 30 MeV and the baryon density range ps < 1014 2 g/cm3. In this region, in addition to the mean-field effects the formation of few-body correlations, in particular light bound clusters up to the alpha-particle (1 < A < 4) has been taken into account. The calculation is based on the relativistic mean field theory with the parameter set TM1. We show results for different values for the asymmetry parameter, and (3 equilibrium is considered as a special case. 

Figure 4 Overview of several theoretical predictions for the SE Brueckner-Hartree-Fock (continuous choice) with Reid93 potential (circles), self-consistent Green function theory with Reid93 potential (full line), variational calculation from [9] with Argonne Avl4 potential (dashed line), DBHF calculation from [16] (triangles), relativistic mean-field model from [22] (squares), effective field theory from [23] (dash-dotted fine).

FM at some density 1. One of the essential points we learned here is that we need no spin-dependent interaction at the original Lagrangian to see SSP. We can see a similar phenomenon in dealing with nuclear matter within the relativistic mean-field theory, where the Fock interaction can be extracted by way of the Fierz transformation from the original Lagrangian [11],... 

If we understand FM or magnetic properties of quark matter more deeply, we must proceeds to a self-consistent approach, like Hartree-Fock theory, beyond the previous perturbative argument. In ref. [11] we have described how the axial-vector mean field (AV) and the tensor one appear as a consequence of the Fierz transformation within the relativistic mean-field theory for nuclear matter, which is one of the nonperturbative frameworks in many-body theories and corresponds to the Hatree-Fock approximation. We also demonstrated... 

Methods for treating relativistic effects in molecular quantum mechanics have always seemed to me, if I may say so without appearing too impertinent to those who work in the field, a complete dog s breakfast. The difficulty is to know to what question they are supposed to be the answer, in the circumstances in which we find ourselves. We do not know what a relativistically invariant theory applicable to molecular behaviour might look like. As was pointed out to us at the last meeting, the Dirac equation certainly will not do to describe interacting electrons and even at the single particle level, where it seems to work, there is an inconsistency in interpreting its solutions in terms... 

Nonrelativistic quantum electrodynamics (NRQED) [11] is an attempt to combine the simplicity of the quantum mechanical description with the power and rigor of field theory. The idea is to write ordinary relativistic quantum electrodynamics in the form of a nonrelativistic expansion with a Lagrangian containing vertices with arbitrary powers of fields. This is useful if we want to consider essentially nonrelativistic processes, like nonrelativistic bound states and threshold phenomena. In such a physical situation the dominant dynamics is nonrelativistic, and the calculations could be in principle simplified if... 

Contributions to the energy which depend only on the small parameters a. and Za. are called radiative corrections. Powers of a arise only from the quantum electrodynamics loops, and all associated corrections have a quantum field theory nature. Radiative corrections do not depend on the recoil factor m/M and thus may be calculated in the framework of QED for a bound electron in an external field. In respective calculations one deals only with the complications connected with the presence of quantized fields, but the two-particle nature of the bound state and all problems connected with the description of the bound states in relativistic quantum field theory still may be ignored. 

This experimental development was matched by rapid theoretical progress, and the comparison and interplay between theory and experiment has been important in the field of metrology, leading to higher precision in the determination of the fundamental constants. We feel that now is a good time to review modern bound state theory. The theory of hydrogenic bound states is widely described in the literature. The basics of nonrelativistic theory are contained in any textbook on quantum mechanics, and the relativistic Dirac equation and the Lamb shift are discussed in any textbook on quantum electrodynamics and quantum field theory. An excellent source for the early results is the classic book by Bethe and Salpeter [6]. A number of excellent reviews contain more recent theoretical results, and a representative, but far from exhaustive, list of these reviews includes [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. 

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