Theoretical Physics, ANL Emeritus. Fellow of the American Physical Society, 2010. Fellow of the Institute of Physics (UK), 2004. Adjunct Professor of physics, University of Miami, 2015. A.B., Magna Cum Laude, Princeton University, 1974. Ph.D. Caltech, 1979 Supervisors: John Schwarz
Cosmas Zachos , Colloquium at the Physics Dept of the University of Miami, Oct 14, 2020. ... more Cosmas Zachos , Colloquium at the Physics Dept of the University of Miami, Oct 14, 2020.
Wigner's 1932 quasi-probability Distribution Function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum flows in: semiclassical limits; quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" puzzles. It is also of importance in signal processing (time-frequency analysis).
Nevertheless, a remarkable aspect of its internal logic, pioneered by J Moyal, and H Groenewold, has only emerged in this millennium: It furnishes a third, alternate, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations, and perhaps more intuitive, since it shares language with classical mechanics.
It is logically complete and self-standing, and accommodates the uncertainty principle in an unexpected manner.
In the quantized two-dimensional nonlinear supersymmetric σ-model, the supercurrent supermultiple... more In the quantized two-dimensional nonlinear supersymmetric σ-model, the supercurrent supermultiplet, which contains the energy-momentum tensor, is transformed by the nonlocal symmetry of the model into the isospin current supermultiplet. This effect incorporates supersymmetry into the known infinite-dimensional yangian deformation symmetry of plain σ-models, leads to precisely the same nontrivial extension of the two-dimensional super-Poincaré group as found previously for the Poincaré group, and thus determines the theory's mass spectrum. A generalization to all higher-order nonlocal charges is conjectured such that their generating function, the so-called “master charge”, has a definite Lorentz spin which depends on the spectral parameter.
The geometric picture of the star-product based on its Fourier representation kernel is utilized ... more The geometric picture of the star-product based on its Fourier representation kernel is utilized in the evaluation of chains of star-products and the intuitive appreciation of their associativity and symmetries. Such constructions appear even simpler for a variant asymmetric product, and carry through for the standard star-product supersymmetrization. §1. Groenewold’s noncommutative ⋆-product [1] of phase-space functions f(x, p) and g(x, p) is the unique associative pseudodifferential deformation [2] of ordinary products: ⋆ ≡ e ← ∂ x → ∂ p− ← ∂ p → ∂ x)/2 . (1) It is the linchpin of deformation (phase-space) quantization [3, 2], as well as applications of matrix models and non-commutative geometry ideas in M-physics [4]. In practice, since it involves exponentials of derivative operators, it may be evaluated through translation of function arguments, f(x, p) ⋆ g(x, p) = f(x + ih̄ 2 → ∂ p, p − ih̄ 2 → ∂ x) g(x, p). (2) However, explicit evaluations of long strings of star-products in...
The Wigner phase-space distribution function provides the basis for Moyal's deformation quant... more The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. General features of time-independent Wigner functions are explored here, including the functional ("star") eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux ("supersymmetric") isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the Poeschl-Teller potential, and the Liouville potential.
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) envelo... more A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
The dynamics of topological open branes is controlled by Nambu Brackets. Thus, they might be quan... more The dynamics of topological open branes is controlled by Nambu Brackets. Thus, they might be quantized through the consistent quantization of the underlying Nambu brackets, including odd ones: these are reachable systematically from even brackets, whose more tractable properties have been detailed before.
In recent years the umbral calculus has emerged from the shadows to provide an elegant correspond... more In recent years the umbral calculus has emerged from the shadows to provide an elegant correspondence framework that automatically gives systematic solutions of ubiquitous difference equations --- discretized versions of the differential cornerstones appearing in most areas of physics and engineering --- as maps of well-known continuous functions. This correspondence deftly sidesteps the use of more traditional methods to solve these difference equations. The umbral framework is discussed and illustrated here, with special attention given to umbral counterparts of the Airy, Kummer, and Whittaker equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg--de Vries, and Toda systems.
Phase Space is the framework best suited for quantizing superintegrable systems, naturally preser... more Phase Space is the framework best suited for quantizing superintegrable systems, naturally preserving the symmetry algebras of the respective hamiltonian invariants. The power and simplicity of the method is fully illustrated through new applications to nonlinear sigma models, specifically for de Sitter N-spheres and Chiral Models, where the symmetric quantum hamiltonians amount to compact and elegant expressions. Additional power and elegance is provided by the use of Nambu Brackets to incorporate the extra invariants of superintegrable models. Some new classical results are given for these brackets, and their quantization is successfully compared to that of Moyal, validating Nambu's original proposal.
In the context of phase-space quantization, matrix elements and observables result from integrati... more In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.
Cosmas Zachos , Colloquium at the Physics Dept of the University of Miami, Oct 14, 2020. ... more Cosmas Zachos , Colloquium at the Physics Dept of the University of Miami, Oct 14, 2020.
Wigner's 1932 quasi-probability Distribution Function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum flows in: semiclassical limits; quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" puzzles. It is also of importance in signal processing (time-frequency analysis).
Nevertheless, a remarkable aspect of its internal logic, pioneered by J Moyal, and H Groenewold, has only emerged in this millennium: It furnishes a third, alternate, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations, and perhaps more intuitive, since it shares language with classical mechanics.
It is logically complete and self-standing, and accommodates the uncertainty principle in an unexpected manner.
In the quantized two-dimensional nonlinear supersymmetric σ-model, the supercurrent supermultiple... more In the quantized two-dimensional nonlinear supersymmetric σ-model, the supercurrent supermultiplet, which contains the energy-momentum tensor, is transformed by the nonlocal symmetry of the model into the isospin current supermultiplet. This effect incorporates supersymmetry into the known infinite-dimensional yangian deformation symmetry of plain σ-models, leads to precisely the same nontrivial extension of the two-dimensional super-Poincaré group as found previously for the Poincaré group, and thus determines the theory's mass spectrum. A generalization to all higher-order nonlocal charges is conjectured such that their generating function, the so-called “master charge”, has a definite Lorentz spin which depends on the spectral parameter.
The geometric picture of the star-product based on its Fourier representation kernel is utilized ... more The geometric picture of the star-product based on its Fourier representation kernel is utilized in the evaluation of chains of star-products and the intuitive appreciation of their associativity and symmetries. Such constructions appear even simpler for a variant asymmetric product, and carry through for the standard star-product supersymmetrization. §1. Groenewold’s noncommutative ⋆-product [1] of phase-space functions f(x, p) and g(x, p) is the unique associative pseudodifferential deformation [2] of ordinary products: ⋆ ≡ e ← ∂ x → ∂ p− ← ∂ p → ∂ x)/2 . (1) It is the linchpin of deformation (phase-space) quantization [3, 2], as well as applications of matrix models and non-commutative geometry ideas in M-physics [4]. In practice, since it involves exponentials of derivative operators, it may be evaluated through translation of function arguments, f(x, p) ⋆ g(x, p) = f(x + ih̄ 2 → ∂ p, p − ih̄ 2 → ∂ x) g(x, p). (2) However, explicit evaluations of long strings of star-products in...
The Wigner phase-space distribution function provides the basis for Moyal's deformation quant... more The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. General features of time-independent Wigner functions are explored here, including the functional ("star") eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux ("supersymmetric") isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the Poeschl-Teller potential, and the Liouville potential.
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) envelo... more A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
The dynamics of topological open branes is controlled by Nambu Brackets. Thus, they might be quan... more The dynamics of topological open branes is controlled by Nambu Brackets. Thus, they might be quantized through the consistent quantization of the underlying Nambu brackets, including odd ones: these are reachable systematically from even brackets, whose more tractable properties have been detailed before.
In recent years the umbral calculus has emerged from the shadows to provide an elegant correspond... more In recent years the umbral calculus has emerged from the shadows to provide an elegant correspondence framework that automatically gives systematic solutions of ubiquitous difference equations --- discretized versions of the differential cornerstones appearing in most areas of physics and engineering --- as maps of well-known continuous functions. This correspondence deftly sidesteps the use of more traditional methods to solve these difference equations. The umbral framework is discussed and illustrated here, with special attention given to umbral counterparts of the Airy, Kummer, and Whittaker equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg--de Vries, and Toda systems.
Phase Space is the framework best suited for quantizing superintegrable systems, naturally preser... more Phase Space is the framework best suited for quantizing superintegrable systems, naturally preserving the symmetry algebras of the respective hamiltonian invariants. The power and simplicity of the method is fully illustrated through new applications to nonlinear sigma models, specifically for de Sitter N-spheres and Chiral Models, where the symmetric quantum hamiltonians amount to compact and elegant expressions. Additional power and elegance is provided by the use of Nambu Brackets to incorporate the extra invariants of superintegrable models. Some new classical results are given for these brackets, and their quantization is successfully compared to that of Moyal, validating Nambu's original proposal.
In the context of phase-space quantization, matrix elements and observables result from integrati... more In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.
Bosonic q-oscillators commute with themselves and so their free distribution is Planckian. In a c... more Bosonic q-oscillators commute with themselves and so their free distribution is Planckian. In a cavity, their emission and absorption rates may grow or shrink---and even diverge---but they nevertheless balance to yield the Planck distribution via Einstein's equilibrium method, (a careless application of which might produce spurious q-dependent distribution functions). This drives home the point that the black-body energy distribution is not a handle for distinguishing q-excitations from plain oscillators. A maximum cavity size is suggested by the inverse critical frequency of such emission/absorption rates at a given temperature, or a maximum temperature at a given frequency. To remedy fragmentation of opinion on the subject, we provide some discussion, context, and references.
Wigner's quasi-probability distribution function in phase space is a special (Weyl-Wigner) repres... more Wigner's quasi-probability distribution function in phase space is a special (Weyl-Wigner) representation of the density matrix. It has been useful in describing transport in quantum optics; nuclear physics; and quantum computing, decoherence, and chaos. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter century: It furnishes a third, alternative formulation of quantum mechanics, independent of the conventional Hilbert space or path integral formulations.
In this logically complete and self-standing formulation, one need not choose sides between coordinate and momentum space. It works in full phase-space while accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but which compose together in novel algebraic ways.
This volume is a selection of 23 classic and/or useful papers about the phase-space formulation, with an introductory overview that provides a trail-map to these papers, and with an extensive bibliography. The overview collects often-used formulas and simple illustrations, suitable for applications to a broad range of physics problems, as well as teaching. It thereby provides supplementary material that may be used for a beginning graduate course in quantum mechanics.
This volume contains the proceedings of the workshop, Gauge
Theory on a Lattice: 1984, held at A... more This volume contains the proceedings of the workshop, Gauge
Theory on a Lattice: 1984, held at Argonne National Laboratory on
April 5-7, 1984. The workshop was a project of the particle physics
theory group and was jointly sponsored by the High Energy Physics
Division and the Department of Educational Programs of the Laboratory.
In the past few years there have been rapid advances In under-
standing quantum field theory by making discrete approximations of
the path Integral functional. This approach offers a systematic
alternative to perturbation theory and opens up the possibility of
first-principles calculation of new classes of observables.
Computer simulations based on lattice regularization have already
provided Intriguing Insights Into the long-distance behavior of
quantum chromodynamics. The objective of the workshop was to bring
together researchers using lattice techniques for a discussion of
current projects and problems- These proceedings aim to communicate
the results to a broader segment of the research community.
We construct a Lagrangian for the massive scalar multiplet, locally invariant under two types of ... more We construct a Lagrangian for the massive scalar multiplet, locally invariant under two types of spinorial transformations (N=2 supersymmetry). Our theory is based on the coupling of the global supermultiplet to N=2 supergravity and corrections generated iteratively in powers of Newton's constant. Consistency of the theory requires the vector field of supergravity to gauge the central charge represented in the massive sector of the multiplet. The same vector may alternatively gauge the internal 0(2) symmetry of the two supersymmetry generators. Furthermore, it may even gauge a linear combination of the generators of these two groups; we indicate the grounds for this compatibility. We discuss the hierarchy of internal symmetries characterizing each sector of the theory, ranging from U(1)xSU(2)xSU(2) down to 0(2)x0(2). This internal symmetry imposes tight constraints on the system. For instance, the nonpolynomial structure of the spinless fields at hand is considerably more restricted than that present in the general simple supersymmetric (N-1) theory. Furthermore, the vector field is forced to couple to the matter fields with gravitational strength, to the effect that the resulting Coulomb potential exactly cancels against the Newtonian potential of gravity, in the static limit. Our theory may be also viewed as a truncation of N=8 supergravity theory, compatibly with the SO (8) breakup scheme into SU(3)xU(1)xU(1). The potential of the spinless fields present has a local minimum at the origin, but further off it is not even bounded from below. However, we point out some indications that the tunneling out of the supersymmetric, metastable vacuum is negligibly small.
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Wigner's 1932 quasi-probability Distribution Function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum flows in: semiclassical limits; quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" puzzles. It is also of importance in signal processing (time-frequency analysis).
Nevertheless, a remarkable aspect of its internal logic, pioneered by J Moyal, and H Groenewold, has only emerged in this millennium: It furnishes a third, alternate, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations, and perhaps more intuitive, since it shares language with classical mechanics.
It is logically complete and self-standing, and accommodates the uncertainty principle in an unexpected manner.
Papers by Cosmas Zachos
Wigner's 1932 quasi-probability Distribution Function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum flows in: semiclassical limits; quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" puzzles. It is also of importance in signal processing (time-frequency analysis).
Nevertheless, a remarkable aspect of its internal logic, pioneered by J Moyal, and H Groenewold, has only emerged in this millennium: It furnishes a third, alternate, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations, and perhaps more intuitive, since it shares language with classical mechanics.
It is logically complete and self-standing, and accommodates the uncertainty principle in an unexpected manner.
In this logically complete and self-standing formulation, one need not choose sides between coordinate and momentum space. It works in full phase-space while accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but which compose together in novel algebraic ways.
This volume is a selection of 23 classic and/or useful papers about the phase-space formulation, with an introductory overview that provides a trail-map to these papers, and with an extensive bibliography. The overview collects often-used formulas and simple illustrations, suitable for applications to a broad range of physics problems, as well as teaching. It thereby provides supplementary material that may be used for a beginning graduate course in quantum mechanics.
Theory on a Lattice: 1984, held at Argonne National Laboratory on
April 5-7, 1984. The workshop was a project of the particle physics
theory group and was jointly sponsored by the High Energy Physics
Division and the Department of Educational Programs of the Laboratory.
In the past few years there have been rapid advances In under-
standing quantum field theory by making discrete approximations of
the path Integral functional. This approach offers a systematic
alternative to perturbation theory and opens up the possibility of
first-principles calculation of new classes of observables.
Computer simulations based on lattice regularization have already
provided Intriguing Insights Into the long-distance behavior of
quantum chromodynamics. The objective of the workshop was to bring
together researchers using lattice techniques for a discussion of
current projects and problems- These proceedings aim to communicate
the results to a broader segment of the research community.