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We discuss in relativistic quantum mechanics the excitation of atoms by photons, assuming an electromagnetic interaction of the form A. It turns out, that when localized in a region V at a certain time x0 a photon would excite atoms also outside the light-cone of V. We arrive thus at a contradiction between quantum mechanics and special relativity.

Using the action principle, we determine the Poincare transformation properties of second-quantized fields in the presence of external fields. We then construct expressions for the generators of the Poincare group and explicitly verify that a second-quantized spin-3/2 field interacting with a minimal external electromagnetic field transforms covariantly. We thus conclude that the negative metric which appears in the quantization of the spin-3/2 field is not the result of a noncovariant quantization as has been claimed.

Angular correlations between the tritons and 16O GS nuclei, produced in the sequential reaction 16O(7Li,t)20Ne*( varies as )16O, have been measured for a number of levels in 20Ne. The tritons were detected at angles as large as 30 degrees in the laboratory and the fits to the correlations suggested the existence of a symmetry axis for the decay of the intermediate state 20Ne. Taking the observed symmetry axis as the axis of quantization the population of the magnetic m=0, +or-1 of levels of 20Ne with known JPi was deduced from the correlations. The data are best fitted by assuming at least 84% population of the m=0 substate. These observations are consistent with the (7Li,t) reaction being mainly a direct reaction, and consequently it can be used reliably to obtain spectroscopic information. The structure of two states at 12.56 and 15.36 MeV is discussed.

§1 lists axioms desirable in any quantization rule for the functions of the q 's. The momentum observables are introduced in §2 prior to their quantization in §4. §5 essentially shows how conventional quantum mechanics fits into this scheme of things. By progressive specialization from a general manifold to a vector space, from a general quantization scheme to one which is linear on the linear momentum functions, and finally to an entirely well-behaved (admissible) quantization rule, we obtain in §7−§9 results which become progressively more and more powerful. The final theorem (Theorem 9.2) is perhaps the most significant of all. This result states that there exists a class of functions which contains all functions of the q 's and functions of the p 's and all momentum observables and which is closed with respect to any linear canonical transformation L ; a rule Λ assigning a unique self-adjoint operator to each such function f ; a unitary operator W L corresponding to L and an equation Λ(f∘L)=W -1 L ΛƒW L . Contents 1. 1. Prequantization schemes 2. 2. Momentum observables 3. 3. The set L ( M ) 4. 4. Quantization schemes 5. 5. An example 6. 6. Systems of imprimitivity 7. 7. Quantization schemes on vector spaces 8. 8. Standard quantization schemes 9. 9. Admissible quantization schemes Appendix References

The operators of the classical electromagnetic field, whose eigenvalues are the corresponding classical fields, are introduced. The use of eigenfunctions of classical field operators in describing the transition from the free quantum field to the classical one is considered. The scheme for the derivation of the semiclassical approximation from nonrelativistic quantum electrodynamics is improved. The limits of applicability of the semiclassical approximation are considered.

The Stuckelberg formalism for a massive spin-one field interacting with external potentials is quantized following the method of Takahashi and Umezawa. It is shown how the characteristic determinant of the field equations, which, in an earlier paper, was shown to have the same form as the Lee-yang determinant, for such a theory, determines the possibility of breakdown of Lorentz invariance in the procedure of the Takahashi-Umezawa quantization method.

The relations betweeb a field F μν M satisfying the usual Maxwell equations and a field F μν D satisfying the symmetric Maxwell-Dirac equations, and the singular potential solving both of these is given. The action principle is formulated in both forms and the reality of the string is shown. A string with spin is constructed by placing electric charges at its end-points. The motion and interactions of the string, the relation between flux and angular momentum quantization and the passage to two-body Hamiltonians are examined.

The Gowdy ${T}^{3}$ inhomogeneous cosmology is an exact vacuum solution of Einstein's equations representing gravitational waves propagating in an expanding closed universe. As shown by Berger, it provides an example showing graviton pair creation as a quantum effect near the initial singularity, together with the reaction this induces in the cosmological expansion. The truncated quantization given here by superspace methods develops the "single residual constraint" quantization method proposed by Moncrief. The quantum wave equation is soluble by separation of variables, and shows that graviton number is a concept unsuited to the description of the initial (velocity-dominated) singularity. Thus a good description of the pair creation occurring in even this soluble model has not yet been developed.

Following previous discussions concerning the field-theoretical derivation of Kadanoff's scaling laws, we apply the method of "soft quantization" to the derivation of a homogeneous renormalization-group equation. This equation is similar to the one proposed recently by S. Weinberg. In addition to our attempt to close the "communication gap" between physicists working on critical phenomena and high-energy physics, we discuss some new applications of such homogeneous differential equations to perturbations of scale-invariant models.

DII~AC (1) demonstrated long ago that the existence of magnetic monopoles together with the formalism of quantum mechanics necessarily implies that the sources of the electromagnetic field (electric and magnetic charge) are quantized. Several different arguments leading to Dirac's conclusion have been put forth, notably those due to SC~WI~G~n (2) and CABIBBO and I~ERRARI (a). The method of the latter two authors has the distinct advantage of relativistic covariance. Recently MOTZ (4) has adapted Schwinger's line of reasoning to general relat ivi ty in an a t tempt to demonstrate the quantization of the source of the gravitational field; that is the quantization of mass. The val idi ty of his procedure is open to some criticism as he employs only an approximate two-body equation of motion in his derivation; in addition, the covariance of his result is not clear. The method of CAmnBO and FERRARI in the case of electromagnetism is equivalent in its conclusions to that of SCHWINGER and, in addition to possessing covarianee, their method does not require the use of the equations of motion of the field sources. Therefore, it is the purpose of this note to investigate their procedure as applied to the gr~vitationM field. To this end consider the Dirac equation for a spinor field in curved space

The Bhabha field with multimass and maximum spin S, a system transforming reducibly under a representation of the homogenous Lorentz group is considered. The field is quantized in the presence of a minimally coupled electromagnetic field. The relation ψ(x)=ψ( x σ ) does not hold for the case of S integer contrary to the half-integer S or lower spin (S

The effect of the quantization of the electron energy levels in a strong transverse magnetic field H on the low-temperature thermoelectric power (TEP) S of a high-purity isotropic semiconductor ( n - type gallium arsenide GaAs) is investigated theoretically. The “electron-diffusion” ( S e ) and “phonon-drag” ( S p ) components of S ( = S e + S p ) are calculated in the extreme quantum limit, when all the electrons in the conduction band are concentrated in the lowest Landau level. The transition to nondegeneracy, which takes place when the bottom of the lowest Landau level is driven through the Fermi level, has a large effect on the variations of S e and S p with magnetic field. The results are illustrated with numerical calculations for n - type GaAs at 4.2 K with 1.2 × 10 16 cm -3 electrons.

Lagrangian densities are introduced for nonlocal field theories, which make the application of the action principle possible. The action principle is then applied to classical and quantum nonlocal field theories. General formulas for conserved densities are derived by use of a generalized variation method, under the assumption of c-number variations. These formulas are also applied to a particular model due to Kristensen and Moller. The charge and energy-momentum vector are thereby, shown to be equal to the expressions given by Pauli, which he derived by other means. This model is also quantized by use of the Yang-Feldman method. But as this method leads to a noncanonical quantization, the above derived quantities are in general no longer conserved. This is due to the fact that the assumption of c-number variations essentially restricts the quantization to a canonical one. The action principle with q-number variations is therefore considered. Thus new integral conserved quastities are derivable. However one also gets a general consistency condition for the above model as well as for all other similar models. The fulfilment of this condition is required by (i) stationarity of the total action and (ii) uniqueness of the integrgal conserved quantities. It is also a necessarymore » condition for the existence of a unitary S operator. The Kristensen-Moller model is shown to violate the above condition, and is therefore not consistent.« less

A novel approach to the quantum mechanical description of phase properties of a one-mode low-intensity radiation field is presented. The procedure is based upon the correspondence between Glauber states and classical waves of fixed phase ϕ and amplitude a, and, in contrast to the conventional quantum mechanical phase concepts, allows to make contact between certain operators ℰm corresponding to the classical quantities exp {imφ} (m = ± 1, ± 2, …) and realistic measurements. In the observation, a large amplification of the microscopic field by means of a linear quantum amplifier plays a decisive role. Since the amplified field is a macroscopic one, the measurement of phase is not problematic, and hence, in particular, average values «exp {imϕ}» can be determined experimentally. The main result of our analysis is that the mean values «exp {imϕ}» are equal to the quantum mechanical expectation values for the operators ℰm with respect to the field before amplification. To obtain this result it became necessary to calculate Glauber's P function for the amplified field in case of an arbitrary state for the initial field.

Using electron tunneling through $n$-type PbTe-oxide-Pb junctions, we have measured some electronic properties of the (100) surface accumulation layer at the PbTe-oxide interface. Three energy levels, resulting from quantization of the electronic motion normal to the surface, are observed in a sample having $n=1.2\ifmmode\times\else\texttimes\fi{}{10}^{18}$ /${\mathrm{cm}}^{2}$. The energies of these three levels are ${E}_{0}=42$ meV, ${E}_{1}=19$ meV, and ${E}_{2}=6$ meV below the conduction band edge. From the magnetotunneling data we obtain ${g}^{*}=29\ifmmode\pm\else\textpm\fi{}3$ and ${m}^{*}=(0.069\ifmmode\pm\else\textpm\fi{}0.008){m}_{0}$ for the ground-state subband. In addition, the Landau levels in the ground-state subband do not cross the Landau levels in the excited-state subband. The interaction between the Landau levels removes the degeneracy at the crossover and a splitting, $\ensuremath{\Delta}E\ensuremath{\approx}4$ meV, has been observed.

Linearized theory suggests that the NUT solution of Einstein's equations corresponds to a source with both mass and dual mass i.e. to a gravitational dyon. This is born out by the striking identity between the Killing operators of the NUT solution and the ‘total angular momentum’ operators of the monopole. On this basis, Misner's periodic time condition is shown to be the analogue of the Dirac quantization, and results from the requirement that the generators integrate to a global Lie group. It is also shown that there are no bound states for a Klein-Gordon field in NUT space provided the field vanishes in the conventional way at the horizon. For this purpose a generalized ‘tortoise coordinate’ is introduced.

It is shown that the standard time‐dependent perturbation theory is inadequate to treat the problem of line narrowing by double‐frequency irradiation in NQR. We show here that a semiclassical treatment which avoids the secular terms provides results in agreement with the radiation field quantization method, which is here extended for the first time to NQR problems. As an application, we consider a hypothetical 14N NQR double irradiation experiment and we determine the conditions which lead to line narrowing.

It is shown that if quantum electrodynamics is assumed to be manifestly gauge independent and path independent then electric charge must be quantized.

Simple expressions for the expectation values of the kinetic and potential energies are derived for a Dunham oscillator. For low vibrational levels, these differ only slightly from results derived utilizing the semiclassical quantization condition, and lead to improved numerical agreement with accurate results for H2.

Relativistic field theories with nonlocal interaction are investigated. This is essentially done in terms of the action principle with a generalized variation method previously developed by the author. That a classical theory of the above type has a canonical structure, at least for the perturbative solutions in terms of the asymptotic free fields, was once shown by Pauli. The in and out fields, for example, are equally good choices of canonical variables. A fundamental quantization in the Heisenberg picture is, therefore, Yang-Feldman-K\"all\'en quantization method. However, since the interacting fields never can be canonical variables, owing to the nonlocality in time, the classical expressions of the Poincar\'e generators (properly symmetrized) are no longer conserved and have no longer the right generator properties in the quantum case for any nonlinear model. There exist, however, new generators which are conserved and which have the right generator properties. These generators may be derived from the action principle by means of $q$-number variations instead of $c$-number variations. However, these generators are not uniquely determined. In particular it is shown that any nonlinear model has an in and an out representation of the generators which are different in terms of the interacting fields. They can, therefore, only be equal because of some particular property of the field equations. Explicit calculations are performed for a particular model due to Kristensen and M\o{}ller (a nonlocalized Yukawa coupling). For this model it is shown that the in and out representations indeed are different (in the fourth order in the coupling constant) even for the charge-symmetric version of the model. (The properties under the discrete transformations $C$, $P$, and $T$ are also considered.) This implies that the field equations do not yield unique quantum solutions and in particular that the solutions with canonical incoming free fields are different from the solutions with canonical outgoing free fields, and none of these solutions render the total action stationary. No meaningful $S$ matrix can therefore be defined. It is also shown that this deficiency cannot be corrected either by restricting the form function or by adding correction terms to the perturbation expansions.

A physical characterisation of classical systems in quantum mechanics is given in terms of the set of ensembles in contrast to the well-known characterisations concerning the effects or observables: A quantum mechanical system is classical if and only if each two decompositions of every ensemble are compatible.

A representation of the position and angular-momentum variables of a quantum-mechanical rigid rotator is obtained from general symmetry principles. The method used is similar to the treatment of spin in the case of the quantum particle.

A new approach to finding a steady foundation of quantum field theory is investigated with the help of the so-called non-standard mathematics. We construct a model of the quan"tum field theory as quantum mechanical system with very much degrees of freedom. The importance of the notion of the models incidental to the formally given quantum field theory and the notion of their physical equivalence are strongly emphasized.

It is shown that any (relativistic, action-at-a-distance theory formulated in terms of a (relativistically invariant) Fokker action, which is invariant under any change of the world-line parameter, also can be given a Lagrangian formulation. The parametrization invariance makes it possible to introduce a common parameter which also can be identified with time. The resulting Lagrangians are nonlocal in time and therefore the applicability of the action principle requires a generalized variation method which is developed here. The resulting generators are expected to generate the corresponding variations in a Poisson-bracket sense, i.e., Hamiltonian formulation of the theory is expected to exist. However, in order to check this it is necessary to (formally) solve the equations of motion, as one needs Poisson brackets for unequal time due to the nonlocality of the generators. But since the equations of motion themselves are nonlocal in time, the Newtonian initial data will not yield a unique solution. Thus, in order to retain the Newtonian degrees of freedom one needs a selection principle as a subsidiary condition. Such a principle has been proposed which states that one has to choose the solution which has a nonrelativistic limit. In the case of a general vector interaction the class of solutions which is obtained by an iterative method (which starts from straight lines) is considered. These solutions are uniquely determined by the Newtonian initial data, but they are applicable only to scattering processes. By choosing the asymptotic straight lines as canonical variables it is explicitly shown that the Poincar\'e generators fulfill the Lie algebra of the Poincar\'e group and that the physical positions transform as Lorentz vectors. It is also shown that the physical positions cannot be chosen as canonical variables. Quantization is discussed and a general argument is put forth which states that the above solutions cannot be quantized by imposing the canonical commutation relations on the canonical variables.

The scheme of quantisation of non-local field theory is formulated. An intermediate regularisation is introduced into the non-local Lagrangian of the classical scalar field in such a way that the procedure of the canonical quantisation leads to the appearance of additional ghost states with indefinite metrics. The ghost states disappear when the regularisation is removed but the propagator of the scalar particle becomes non-local and theS-matrix is finite, unitary, causal and covariant in each perturbation order.

The Euclidean-Markov field, extensively used in constructive quantum field theory, is shown to be the same as the lowest energy generalized stochastic process associated with classical field theory through the procedure of Nelson's stochastic mechanics. As a result, the underlying four-dimensional manifold, on which the Markov field is defined, can be considered as the physical space-time.

The author considers the Gel'fand-Yaglom equations for free fields of arbitrary spin, in the particular case when the field transforms according to a direct sum of inequivalent irreducible finite representations of the proper Lorentz group. Under the assumption that the theory carries neither physical states of zero charge or energy density and that the mass-spin states are non-degenerate, he obtains the precise forms of the minimal and characteristic polynomials of the s blocks of the L0 matrix, which are then used to obtain new necessary and sufficient conditions that the theory be quantizable. The representation according to which the field transforms can be depicted graphically in a simple way and he takes advantage of this to use some simple ideas of graph theory to obtain results. The conclusion is that it will probably be necessary to allow repeated irreducible representations of the proper Lorentz group for theories of spin greater than eight to be quantizable.

A spinor field of subcanonical dimension 1/2 appearing as constituent field in a scale- and gauge-invariant nonlinear spinor field theory can be imbedded in a conventional Lagrange field theory with third-order derivatives. In this framework and by restricting the investigations to the corresponding «free» case the «currents», in particular the current proper, the energy-momentum tensor, the spin tensor and the dilatation current, can be constructed in classical analogy and quantization can be introduced along conventional lines. The theory can be cast into canonical form by introducing two additional spinor fields which are related to first- and second-order derivatives of the original field by Euler equations. The subcanonical spinor field theory appears as a certain projection of the canonical theory which has to be interpreted as a projection on the underlying state space in order to preserve the algebraic relations. The currents and as a consequence their commutators are not uniquely determined by their generator properties. To identify them with physical currents, additional conditions connected with the positive-definite metric of the physical subspace have to be required which single out the formal «Noether currents» of the canonical version of the theory as the only physically acceptable ones. As a consequence the leading singularities of the current commutator on the light-cone are found to agree with those of the Dirac, theory whereas the disconnected part contains a factor of three.

A series of experiments were performed to determine the effect of error signal quantization on human operator compensatory tracking performance. Single-axis, dual-axis, and cross-coupled critical tracking tasks were utilized with a variety of quantization formats. The controlled element dynamics were chosen so as to force the operator to generate varying amounts of lead equalization. The single-and dual-axis critical tasks served as sensitive indicators of display format effects, with the single-axis task yielding information concerning the operator's effective time delay while tracking. The cross-coupled task allowed measurement of the operator's attentional workload margin while using the quantized displays. The results indicate increased operator time delays and attentional workload when using the quantized display formats.

The c-number variation in the quantum-mechanical action principle of Schwinger is extended to a q-number variation which uses an action integral analogous to the classical modified Hamilton action integral. An explicit realization of the admissible q-number variation for the hamiltonian operator, H, given by H(q,p,t)=1/2pjgjk(q)pk+1/2(Aj(q),pj)+W(q,t) is discussed in terms of Gauteaux variation. The action principle yields the correct Hamilton-Heisenberg equations and a relationship between the lagrangian L and the hamiltonian H. It is also shown that while the fundamental commutation relation is successfully derived by Schwinger in his c-number variational principle, the same argument cannot be used in the present q-number variation. A new method of quantization is suggested.

From the infinite order equations of motion of conventional electrodynamics, one can extract by order depression a subclass of second order equations of motion parametrized only by initial positions and velocities. This article presents, with a view toward possible later quantization, a canonical formulation of this electrodynamics. It happens to have the same aspect as for free particles: H=(m12+p12)1/2 +(m22+p22)1/2. The p's are constant, and the canonical variables q's describe straight lines (particle positions cannot be canonical). The extension to the many‐body problem is given.

Beginning with free field quantization on hyperboloids in the forward lightcone, we extend field expansions to the full space‐time. The role of boundary conditions on the propagator and of the particle‐antiparticle distinction in establishing a unique field expansion is discussed. Some difficulties with the possibility of a surface to surface development of the S matrix are encountered in the massless case. Even in the massive case hyperboloids are unsuitable quantization surfaces when x2

An extended Bohr model is proposed, incorporating Landau’s quantiza­tion, for the treatment of magnetic splittings of atomic levels at high quantum numbers. The results agree well with the splittings observed by Garton & Tomkins (1969) in the 6s 2 1 S 0 –6snp 1 P 0 1 series of BaI for values of n up to 32, and qualitatively up to n = 40.

The second-order WKB quantization condition is used to derive expressions for the second-order contributions to the vibrational energy levels and the rotational and centrifugal constants of a diatomic molecule. The second-order corrections to the RKR (Rydberg-Klein-Rees) turning points are also calculated. These second-order quantities are related to the quantities calculated from Kaiser's modified first-order quantization condition. Numerical calculations for the ground electronic state of carbon monoxide are compared with quantum-mechanical values obtained by numerical solution of the Schrodinger equation. In general, the second-order semiclassical values are in good agreement with the quantum-mechanical values, and can be computed by the present methods in a small fraction of the time.

Attempts to obtain a quantum theory of gravitation are reviewed. The essential ideas of those programmes on which extensive work has been done are discussed. Successes and difficulties of each of these programmes are described. The attempts are divided into two broad classes: those based on the techniques of canonical quantization and those based on the techniques of quantum field theory. The authors approach is rather geometrical, an approach which facilitates the comparison of various programmes.

Lagrangian theories involving higher-order derivatives are reconsidered. Such theories appear of great interest as an imbedding for field theories with fields of subcanonical dimension, likee.g. the Heisenberg nonlinear spinor theory, for which local interactions are less singular than in the canonical case. The conservation laws derived according to Noether and Bessel-Hagen are stated in a compact form. Quantization is introduced without making use of the canonical formalism.