• N Mukunda

      Articles written in Pramana – Journal of Physics

    • Development of the analogy between classical and quantum mechanics

      N Mukunda

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      A quantum-mechanical generalisation of Carathéodory’s theorem in classical dynamics is established. Several related properties of classical canonical transformations are also generalised to the quantum case.

    • Properties of the symplecton calculus

      N Mukunda

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      The representation of the group SU(2) afforded by the symplecton calculus of Biedenharn and Louck is mathematically related to the older treatments of the representations of this group. The method used is similar to the phase space description of quantum mechanics, and considerably simplifies important calculations.

    • The Hamilton-Jacobi equation revisited

      K Babu Joseph N Mukunda

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      A new analysis of the nature of the solutions of the Hamilton-Jacobi equation of classical dynamics is presented based on Caratheodory’s theorem concerning canonical transformations. The special role of a principal set of solutions is stressed, and the existence of analogous results in quantum mechanics is outlined.

    • New light on the optical equivalence theorem and a new type of discrete diagonal coherent state representation

      N Mukunda E C G Sudarshan

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      In many instances we find it advantageous to display a quantum optical density matrix as a generalized statistical ensemble of coherent wave fields. The weight functions involved in these constructions turn out to belong to a family of distributions, not always smooth functions. In this paper we investigate this question anew and show how it is related to the problem of expanding an arbitrary state in terms of an overcomplete subfamily of the overcomplete set of coherent states. This provides a relatively transparent derivation of the optical equivalence theorem. An interesting by-product is the discovery of a new class of discrete diagonal representations.

    • Algebraic aspects of the wigner distribution in quantum mechanics

      N Mukunda

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      The algebraic structure underlying the method of the Wigner distribution in quantum mechanics and the Weyl correspondence between classical and quantum dynamical variables is analysed. The basic idea is to treat the operators acting on a Hilbert space as forming a second Hilbert space, and to make use of certain linear operators on them. The Wigner distribution is also related to the diagonal coherent state representation of quantum optics by this method.

    • Relativistic particle interactions—A comparison of independent and collective variable models

      J Samuel N Mukunda

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      We present a detailed comparison of two models for relativistic classical particle interactions recently discussed in the literature—one based on independent particle variables, and the other on centre of mass plus relative variables. Basic to a meaningful comparison is a reformulation of the latter model which shows that it makes essential use of the concept of invariant relations from constrained Hamiltonian theory. We conclude that these two models have very different physical and formal structures and cannot be thought of as two equivalent descriptions of the same physical theory.

    • Group theoretical methods in optics

      N Mukunda

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      Scalar Fourier optics is concerned with the passage of paraxial light beams through ideal optical systems. It is well known that the action of the latter on the former can be given in the framework of the two- and four-dimensional real symplectic groups. It is shown here that, based on an analysis of the Poincaré symmetry of the complete Maxwell equations in the front form, a natural representation for paraxial Maxwell beams emerges, which moreover shows the way to a generalization of scalar to vector Fourier optics preserving the group structure of ideal optical systems. Properties of generalized rays, and the usefulness of some pseudo-orthogonal groups in the treatment of Gaussian Schell-model beams, are also brought out.

    • The three faces of Maxwell’s equations

      N Mukunda E C G Sudarshan

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      In dealing with electromagnetic phenomena and in particular the phenomena of optics, despite the recognition of the quanta of light people tend to talk of the amplitudes and field strengths, as if the electromagnetic field were a classical field. For example we measure the wavelength of light by studying interference fringes. In this paper we study the interrelationship of three ways of looking at the problem: in terms of classical wave fields, wave function of the photon; and the quantized wave field. The comparison and contrasts of these three modes of description are carefully analyzed in this paper. The ways in which these different modes complement our intuition and insight are also discussed.

    • Group representations and the method of sections

      Biswadeb Dutta N Mukunda

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      The unitary representations of the Euclidean and Poincaré groups are analysed, using viewpoints suggested by the method of sections, as applied to the monopole problem, and by the method of induced representations.

    • Hamilton’s theory of turns and a new geometrical representation for polarization optics

      R Simon N Mukunda ECG Sudarshan

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      Hamilton’s theory of turns for the group SU(2) is exploited to develop a new geometrical representation for polarization optics. While pure polarization states are represented by points on the Poincaré sphere, linear intensity preserving optical systems are represented by great circle arcs on another sphere. Composition of systems, and their action on polarization states, are both reduced to geometrical operations. Several synthesis problems, especially in relation to the Pancharatnam-Berry-Aharonov-Anandan geometrical phase, are clarified with the new representation. The general relation between the geometrical phase, and the solid angle on the Poincaré sphere, is established.

    • The real symplectic groups in quantum mechanics and optics

      Arvind B Dutta N Mukunda R Simon

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      We present a utilitarian review of the family of matrix groups Sp(2n, ℛ), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of Sp(2n, ℛ). Global decomposition theorems, interesting subgroups and their generators are described. Turning ton-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U(n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n, ℛ) action are delineated.

    • Relativistic operator description of photon polarization

      Arvind N Mukunda

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      We present an operator approach to the description of photon polarization, based on Wigner’s concept of elementary relativistic systems. The theory of unitary representations of the Poincarè group, and of parity, is exploited to construct spinlike operators acting on the polarization states of a photon at each fixed energy momentum. The nontrivial topological features of these representations relevant for massless particles, and the departures from the treatment of massive finite spin representations are highlighted and addressed.

    • The quantum geometric phase as a transformation invariant

      N Mukunda

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      The kinematic approach to the theory of the geometric phase is outlined. This phase is shown to be the simplest invariant under natural groups of transformations on curves in Hilbert space. The connection to the Bargmann invariant is brought out, and the case of group representations described.

    • The algebra and geometry ofSU(3) matrices

      K S Mallesh N Mukunda

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      We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular groupSU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real linear vector space, are developed in anSU(3) covariant manner. Thef andd symbols ofSU(3) lead to two ways of ‘multiplying’ two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization ofSU(3) is developed as a generalization of that forSU(2), and the specifically new features are brought out. Application to the dynamics of three-level systems is outlined.

    • Operator properties of generalized coherent state systems

      N Mukunda

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      The main properties of standard quantum mechanical coherent states and the two generalizations of Klauder and of Perelomov are reviewed. For a system of generalized coherent states in the latter sense, necessary and sufficient conditions for existence of a diagonal coherent stable representation for all Hilbert-Schmidt operators are obtained. The main ingredients are Clebsch-Gordan theory and induced representation theory.

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