Volume 6, Issue 10, October 1965
Index of content:

Microcanonical Ensemble in Quantum Statistical Mechanics
View Description Hide DescriptionThe infinite‐volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamicentropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.

Magnetic Properties of Charged Ideal Quantum Gases in n Dimensions
View Description Hide DescriptionWe consider the mathematical model of an ideal gas of charged bosons or fermions in an n‐dimensional space, treating n as a continuous variable. The investigation shows the extent to which the magnetic behavior depends on the dimensionality of the system. In particular, the charged Bose gas in a homogeneous magnetic field does not condense unless n > 4, in contrast to the field‐free gas which condenses for n > 2: however so long as n > 2 and T < T _{c} this system completely expels homogeneous magnetic fields weaker than a certain critical field,H _{c}. (T _{c} is the field free transition temperature.) This field expulsion comes explicitly from the condensed ground‐state bosons for n > 4; but it is still present, and H _{c} has the same form, for 4 > n > 2 where there is no condensation.

A Proof of Nakanishi's Inequality
View Description Hide DescriptionNakanishi's conjectured inequality for the coefficients of the external masses and invariants in the Feynman denominators for scattering processes is proved by a determinantal method suggested by the analogy between Feynman diagrams and electrical networks.

Mixing of Regge Poles and Cuts in a Field‐Theory Model
View Description Hide DescriptionThe Regge poles generated by ladder diagrams in a λ φ^{3}theory are mixed with the moving cuts generated by a class of nonplanar graphs containing an internal ladder. Since the contribution from the cut‐generating diagram is of order t ^{−2}ln ^{m}t, where t is the asymptotic variable, the t ^{−2} behavior of the pure ladder graphs is examined and the trajectory of the Regge pole near = −2 is calculated. The Mellin transform method is used throughout. The transformed amplitude corresponding to a single cut insertion is given by a product of the form pole‐cut‐pole, where it is only the second Regge trajectory that mixes with the cut. The cut itself depends on the leading trajectory. This result substantiates the predictions as to form of other work based on unitarity, but differs in that the cut and pole depend on different trajectory functions. Finally, multiple insertions of the cut diagrams are shown to generate an amplitude with two moving poles on each sheet of the cut.

A Covariant Theory of Relativistic Brownian Motion I. Local Equilibrium
View Description Hide DescriptionThis paper is devoted to the study of some relativistic methods in statistical mechanics and their applications to the construction of a covariant theory of Brownian movement. In this work we deal only with times which are long compared with the relaxation time β^{−1}. As a consequence of this covariant theory we derive relativistic Wiener integrals.

On the Solution of the BBGKY Equations for a Dense Classical Gas
View Description Hide DescriptionFormal solutions of the BBGKY equations of classical statistical mechanics are obtained in the form of integral equations. This form makes it particularly straightforward to obtain the density corrections to the functional equation for f _{2} and to the evolution equation for f _{1}. The physical implications of the derivation, in particular the validity of Bogoliubov's functional hypothesis, are discussed in detail.

Variational Principle for the Time‐Dependent Schrödinger Equation
View Description Hide DescriptionIn this paper a variational principle is established which fully characterizes the initial‐value problem associated with the time‐dependent Schrödinger equation.

Hydromagnetic Switch‐on Shocks
View Description Hide DescriptionIn one‐dimensional hydromagnetic flow, switch‐on shocks have been shown to be nonevolutionary, e.g., by Syrovatskii. Such shocks, however, may be considered as a superposition of a switch‐on shock and an Alfvén shock, and in this paper some results of such an assumption are examined. If the two shocks remain superimposed, the shock pair is, in fact, evolutionary. If the pair splits under the influence of incoming waves, a first‐order nonlinear analysis indicates that highly unstable situations can occur. But there is a non‐uniqueness implicit in the ansätz, and the author has not been able to exclude the possibility that a solution always exists which is stable except between shocks.

On Some Unitary Representations of the Galilei Group I. Irreducible Representations
View Description Hide DescriptionThe true irreducible unitary representations of central extensions G_{M} of the Galilei universal covering group G and hence the physical representations of G are constructed by Mackey's method of induced representations. The elements of the representation space H are obtained from functions defined on G_{M} and restricted to their values at one representative of each left coset of G_{M} modulo K where K is the induction subgroup. The physical interpretation of these functions is in terms of wave functions and comes from the definition of a basis in H. This interpretation depends on the choice of a fundamental frame of reference in space‐time and on the physical meaning given to a fundamental state. To a change of the representatives corresponds a change of basis in H. By a suitable choice of these representatives, we obtain in particular the momentum‐spin representation and the momentum‐helicity representation. The zero mass case named class II by Inönü and Wigner is then obtained by the limit process M → 0 applied to the helicity representation.

Inequalities for the Solutions of Linear Integral Equations
View Description Hide DescriptionInequalities are derived for the solutions of linear integral equations of a certain class in terms of their inhomogeneous terms and kernels. The construction of these inequalities appears to be very simple in practice as it involves only quadratures. The results presented here are, therefore, expected to be useful in the investigation of physical problems.

Branching Rules for Simple Lie Groups
View Description Hide DescriptionIf Γ is an irreducible representation of a group G, and H is a subgroup of G, then Γ furnishes a representation of H which is, in general, reducible, and the branching rules specify which irreducible representations of H occur in the decomposition of this representation. Branching rules are derived for various choices of G and H, including most possibilities that have been discussed as higher symmetry groups.

Recursion Relations for the Wigner Coefficients of Unitary Groups
View Description Hide DescriptionThe polynomials in the components of a set of n‐dimensional vectors that form a basis for an irreducible representation of SU_{ n } are shown to be part of the basis of the group U_{ nr }, in which the subgroup U_{ n } × U_{ r } is explicitly reduced and r ≥ n ‐ 1. Using this result, the concept of auxiliary Wigner coefficient is introduced, for which the problem of multiplicity does not arise and the phase convention is related to that of Gel'fand and Zetlin; recursion relations for this auxiliary coefficient are obtained in a straightforward way, and the connection between it and the ordinary Wigner coefficient is shown to be simple. The recursion relations are being programmed for an electronic computer to allow the systematic evaluation of the Wigner coefficients of SU_{3} and SU_{4}.

Algebraic Tabulation of the Clebsch‐Gordan Coefficients for Reduction of the Product (λ, μ) (3, 0) of Irreducible Representations of SU(3)
View Description Hide DescriptionTables of Clebsch‐Gordan coefficients of SU(3) for the reduction of the product (λ, μ)(3, 0) of representations of SU(3) are constructed by use of the tensor method. Derivations of some crossing and symmetry relations for the SU _{3} Clebsch‐Gordan coefficients are given, and the Clebsch‐Gordan coefficients for the product (μ, λ) (0, 3) are related to those mentioned above. The phase convention used in compiling the tables is stated and explained.

Formalization of the Lagrangian, the Hamiltonian, and Related Concepts
View Description Hide DescriptionWe consider linear system of finite dimensionality in which the system matrix is a well‐behaved function of the independent variable—z or t. The system is assumed to obey a conservation law with zero signature, but is otherwise unrestricted. Using what we call the ``vectorial derivative,'' it is shown that we can obtain Lagrangian and Hamiltonian functions in terms of which the system behavior is described in the usual way. Thus, the methods of classical physics are made available for the study of such systems, without the physical connotations usually carried by such functions.
In carrying out this formalization, we define a functional—e.g., the Hamiltonian—and, at the same time, define the operator that generates the function. It is shown that the interrelation of two functionals involves the appropriate commutator of the operators involved. For example, the Poisson bracket of a functional, F, and the Hamiltonian, H, is the functional obtained from the operator Γ[ΓH, ΓF], whereand F and H are the operators involved in F and H. The analogy to quantum mechanics is striking.
Finally, it is shown that the contact transformations that preserve the Hamiltonian operator form a rotational system in the Lie algebra,L, in which ΓH is embedded. That is, they obey equations of the form M = [ΓH, M]. It follows, then, that all H‐preserving transformations can be split into components in the intersection of the Γ‐unitary group and either L or a subspace derived from L.

N‐Particle Kinematics and Group‐Theoretical Treatment of Phase Space I. Nonrelativistic
View Description Hide DescriptionThis paper is a group‐theoretical study of the kinematics of n nonrelativistic particles. A systematic method is given to construct a new complete set of commuting observables. The method is based on the existence of a group (the ``great group'') which acts transitively on the phase‐space manifold and preserves the phase‐space volume element; the observables are then Casimir operators of the great group and of some of its subgroups, including the usual three‐dimensional rotation group.
Among these collective observables, in addition to the total angular momentum, the most interesting is the ``togetherness operator'' which describes the simultaneous localization of the n particles. This operator is a generalization to n > 2 of the square of orbital angular momentum; its use allows to generalize to n particles the familiar centrifugal barrier arguments.

Three‐Particle Nonrelativistic Kinematics and Phase Space
View Description Hide DescriptionThe kinematics of a nonrelativistic three‐particle system is studied with the help of the general method devised by Lévy‐Leblond and Lurçat. Basis states are constructed which are eigenstates, in addition to the total momentum‐energy, angular momentum, etc., of new observables; among these, the ``togetherness tensor'' describes the simultaneous localization of the three particles and therefore is of great physical interest. All of these observables arise as Casimir operators of a ``great group'' acting on the three‐particle phase‐space manifold in a transitive way, and of some of its subgroups. In the present case, by trying to keep all the particles on the same footing (``democracy'' arguments), we are led to choose the SU _{3} group as a particularly convenient ``great group''. We thus recover completely the Dragt classification of non‐relativistic three‐particle states. The explicit calculation of the basis functions is done in a new way, by analytical methods, solving partial derivative equations. This enables us to establish the most general form of these basis functions.

S‐Matrix Theory and Double Scattering
View Description Hide DescriptionIn the double scattering case, we show that the introduction of a pole in the S‐matrix elements, is consistent with a description in terms of successive interactions.
We then show that among Olive's class of possible ``propagators'' c(k ^{2} ‐ m ^{2} + iε)^{−1} ‐ (1 ‐ c) ·(k ^{2} ‐ m ^{2} ‐ iε)^{−1}, only the usual value c = 1 gives a consistent result.

Racah Algebra for an Arbitrary Group
View Description Hide Description3j‐ and 6j‐symbols are studied for a general group without assuming that the group is ambivalent or multiplicity‐free. The choice of multiplicity label r that distinguishes the equivalent irreducible representations that may arise in a Kronecker product of irreducible representations is left arbitary and no special choices of phase are made. It is found that the 3j‐ and 6j‐symbols obtained have essentially the same properties as the familiar 3j‐ and 6j‐symbols for the rotation group in three dimensions.

Time‐Dependent Perturbation Theory
View Description Hide DescriptionA time‐dependent perturbation theory which is based upon the U‐matrix approach is presented using the interaction e ^{‐αt }(1 + e ^{‐βt })^{−1} V(x) with β real, α complex and β > Re α > 0. Thus the adiabatic or time‐independent approximation (β → 0) and the sudden approximation (β → ∞) can be obtained using just one formalism. The usual series for the U matrix is derived and shown to converge for a semi infinite range if the interaction is of the form t^{‐δ}, δ > 1, for large times t. Two interesting results are (1) The derivation of the ``golden rule'' from the discrete‐state time‐dependent perturbation theory presented in most text books of quantum mechanics leads to erroneous physical interpretations because the energy‐level shift is ignored. (2) When considering scattering between states in the continuum, it is found that a characteristic feature of time‐dependent interactions is a discrete momentum spectrum of final states which, in the relativistic case, leads to a mass spectrum. These spectra cannot be obtained using the S matrix.