Table of contents

This volume contains a unique collection of papers contributed by experts with a long association with George Sudarshan in seven different areas of physics. Each paper recognizes the seminal contributions to physics made by George Sudarshan, and acknowledges their impact on their own research. They were presented at the conference Sudarshan: Seven Science Quests held at the University of Texas at Austin in November 2006. Participants spanned George's 5 decades of research and represented three of the 4 inhabited continents. Our only regret is that due to the vast breadth of George's quests, some topics (e.g. Quantum Field Theory), had to be left out. The conference was truly a celebration of Physics itself, due to the participation and contributions of a galaxy of stellar physicists who are leaders in their chosen fields.

The focal point of the conference was to showcase George Sudarshan's breakthrough initiation of these seven areas of physics, and to celebrate his 75th birthday. Many of his former and current students, colleagues, friends and family gathered in Austin, Texas for 2 days and presented their research that was based on his prior and current works, and beautiful personal memories.

Papers presented traced the history of the origin of the seven physics fields in which George's role as the originator/initiator has often been overlooked or misrepresented, and the status of these fields in current times. Even many of his colleagues who have worked with him on one or two of the quests were unaware of the impressive breadth and depth of his contributions to the other quests. We feel that the conference achieved its objective of creating a unified showcase of some of his major contributions in a single volume. Of course, knowing George, his quests are ongoing. We wish him many more years of productive pursuit of science.

This meeting would not have been possible without the financial support of several institutions. We are deeply indebted to the Office of the Vice President of Research, the Physics Department, and the Center for Electromagnetic Materials and Devices of The University of Texas at Austin; and to MetaMaterials LLC, Austin,Texas.

Finally, we would like to thank the participants, and particularly George's family, for their contribution to the wonderful atmosphere achieved during the Conference. We would like also to acknowledge the authors of the papers collected in the present volume, the members of the Scientific Committee for their guidance and support, and the referees for their generous work.

A survey of Sudarshan's works is presented to show that his brilliant role in advancing science on many fronts has been like a catalytic agent which, by definition, plays a singular role in increasing the rate of a process, but remains hidden, and is unchanged at the process end. As might be necessary to remain catalytically active for a lifetime, his broad and seminal contributions to science have often gone unappreciated, and rarely given due honor – unfortunately at a great cost to scientific progress.

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I shall discuss some works done by Sudarshan in the 1950's at Rochester. Especially, the topic of magnetic moments of Σ-baryons as well as their electromagnetic mass differences was explained with some historical background. Also, works on weak interaction theory after the discovery of the V-A theory are discussed.

In this talk I will review the V -A theory within the context of the prevalent experimental results at the time.

How V-A has survived for so long despite many a threat to its survival is described with a somewhat idiosyncretic choice of topics.

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We develop an approach to quantum interference phenomenon which makes the characteristics of the problem more conspicuous and brings out more physical concepts. We do this by discussing the idea of deterministic quantum experiments using the Heisenberg picture. The variables that are relevant for describing interference are modular variables. These variables obey non-local equations of motion and can be used to follow the time evolution of a system without disturbing it. We present a physical explanation for the different behaviors of a single particle when the distant slit is open or closed: instead of having a quantum wave that passes through all slits, we have a localized particle with non-local interactions with the other slit(s). While the Heisenberg picture and the Schrodinger pictures are equivalent formulations of quantum mechanics, nevertheless, the results discussed here support a new approach to quantum mechanics which has led to new insights, new intuitions, new experiments, and even the possibility of new devices that were missed from the old perspective.

The study of the symmetries of nature has fascinated scientists for eons. The application of the formal mathematical description of symmetries during the last century has produced many breakthroughs in our understanding of the substructure of matter. In this talk, a number of these advances are discussed, and the important role that George Sudarshan played in their development is emphasized

We review two areas of my work that were directly and indirectly initiated and inspired by George. One is the proofs of no-go theorems in combining the spacetime and internal symmetries in non-trivial ways and the other is how Georges firm conviction on the fundamentality of the spin-statistics theorem helped to expand the domain of applicability of the spin-statistics theorem into the arena of quarks and gluons, going far beyond the original application of the exclusion principle in atomic physics.

In order to provide deeper understanding of mass differences of particles belonging to spin-degenerate multiplets, attempts have been made to see if some non-trivial way of embedding the Lorentz group and internal symmetry groups such as SU(2) and SU(3) into a larger group. When a hint of no-go theorem (that such non-trivial embedding cannot be achieved) first appeared, George went to work and led many of us, including myself, into this area of research. A series of proofs of no-go theorems by George, myself and others eventually led to the definitive proof by Lochlainn O'Raifeartaigh that came to be known as the ORaifeartaigh theorem. Lochlainn also joined George at Syracuse at the same time I went there.

The second area in which George had significant influence on my work is his fervent belief in the fundamental importance of the spin-statistics relationship. First postulated by Worlfgang Pauli to explain the periodic table of elements, the relationship the exclusion principle has exceeded far beyond its original domain of validity. The relationship has been upheld across the scale molecular, atomic, and nuclear structures. What is less known is the fact the relationship continues to remain valid in scales smaller than nucleons. The spin-statistics relationship was one of the compelling reasons for Nambu and I to introduce a new set of then undiscovered degrees freedom for quarks inside nucleons. This new degrees of freedom came to be called the color charges of quarks and it led to the non-Abelian gauge field theory of the strong interactions known today as the quantum chromodynamics. Georges insight into the spin-statistics relationship was indispensable in the birth of the new theory of the strong force.

We review work initiated and inspired by Sudarshan in relativistic dynamics, beam optics, partial coherence theory, Wigner distribution methods, multimode quantum optical squeezing, and geometric phases. The 1963 No Interaction Theorem using Dirac's instant form and particle World Line Conditions is recalled. Later attempts to overcome this result exploiting constrained Hamiltonian theory, reformulation of the World Line Conditions and extending Dirac's formalism, are reviewed.

Dirac's front form leads to a formulation of Fourier Optics for the Maxwell field, determining the actions of First Order Systems (corresponding to matrices of Sp(2,R) and Sp(4,R)) on polarization in a consistent manner. These groups also help characterize properties and propagation of partially coherent Gaussian Schell Model beams, leading to invariant quality parameters and the new Twist phase.

The higher dimensional groups Sp(2n,R) appear in the theory of Wigner distributions and in quantum optics. Elegant criteria for a Gaussian phase space function to be a Wigner distribution, expressions for multimode uncertainty principles and squeezing are described.

In geometric phase theory we highlight the use of invariance properties that lead to a kinematical formulation and the important role of Bargmann invariants. Special features of these phases arising from unitary Lie group representations, and a new formulation based on the idea of Null Phase Curves, are presented.

A seed George planted 45 years ago is still producing fruit now. In 1961, George set out the fundamental proposition that quantum dynamics is described most generally by linear maps of density matrices. Since the first sprout from George's seed appeared in 1962, we have known that George's fundamental proposition can be used to derive the linear Schrodinger equation in cases where it can be expected to apply. Now we have a proof of George's proposition that density matrices are mapped linearly to density matrices, that there can be no nonlinear generalization of this. That completes the derivation of the linear Schrodinger equation. The proof of George's proposition replaces Wigner's theorem that a symmetry transformation is represented by a linear or antilinear operator. The assumption needed to prove George's proposition is just that the dynamics does not depend on anything outside the system but must allow the system to be described as part of a larger system. This replaces the physically less compelling assumption of Wigner's theorem that absolute values of inner products are preserved.

The history of this question is reviewed. Nonlinear generalizations of quantum mechanics have been proposed. They predict small but clear nonlinear effects, which very accurate experiments have not seen. This begs the question. Is there a reason in principle why nonlinearity is not found? Is it impossible? Does quantum dynamics have to be linear? Attempts to prove this have not been decisive, because either their assumptions are not compelling or their arguments are not conclusive. The question has been left unsettled. The simple answer, based on a simple assumption, was found in two steps separated by 44 years.

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After some brief historical remarks we recall the first demonstration of the Spin-Statistics theorem by Pauli, and some further developments, like Streater-Wightman's, in the axiomatic considerations. Sudarshan's short proof then follows by considering only the time part of the kinetic term in the lagrangian: the identity SU(2)=Sp(1,C) is crucial for the argument. Possible generalization to arbitrary dimensions are explored, and seen to depend crucially on the type of spinors, obeying Bott's periodicity 8. In particular, it turns out that for space dimentions (7, 8, or 9) modulo 8, the conventional result does not automatically hold: there can be no Fermions in these dimensions.

This poem was inspired by George Sudarshan's work.

George Sudarshan was quite a bit ahead of the times when he sketched out a non-relativistic proof of the spin-statistics connection forty years back. A recent revival of interest in the role of relativistic arguments in proving the spin-statistics connection prompted us to revisit Sudarshan's original arguments and present them in simpler fashion. The proof is designed to work in three space dimensions. Sudarshan's observation was that if we impose SU(2) invariance, along with a few other restrictions, on the kinematic part of the Lagrangian of a field theory which is derivable from a Weiss-Schwinger type of the Principle of Least Action, then the spin-statistics connection automatically follows. The assumptions and simplifications that go into constructing the proof will be carefully laid out so as to identify the limitations of this approach.

Opening remarks at the Quantum Coherence session.

A review is presented of the early developments in quantum optical coherence. Some properties of the coherent states particularly their over completeness which led to the discovery of diagonal coherent state representation by Sudarshan will be discussed. We then consider some of the important applications of this diagonal coherent state representation.

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A quantum Zeno dynamics can be obtained by means of frequent measurements, frequent unitary kicks or a strong continuous coupling and yields a partition of the total Hilbert space into quantum Zeno subspaces, among which any transition is hindered. We focus on the "continuous" version of the quantum Zeno effect and look at several interesting examples. We first analyze these examples in practical terms, towards applications, then propose a novel experiment.

As of October 2006, there were approximately 535 citations to the seminal 1977 paper of Misra and Sudarshan that pointed out the quantum Zeno paradox (more often called the quantum Zeno effect). In simple terms, the quantum Zeno effect refers to a slowing down of the evolution of a quantum state in the limit that the state is observed continuously. There has been much disagreement as to how the quantum Zeno effect should be defined and as to whether it is really a paradox, requiring new physics, or merely a consequence of "ordinary" quantum mechanics. The experiment of Itano, Heinzen, Bollinger, and Wineland, published in 1990, has been cited around 347 times and seems to be the one most often called a demonstration of the quantum Zeno effect. Given that there is disagreement as to what the quantum Zeno effect is, there naturally is disagreement as to whether that experiment demonstrated the quantum Zeno effect. Some differing perspectives regarding the quantum Zeno effect and what would constitute an experimental demonstration are discussed.

Quantum mechanics enables the calculation of the time evolution of a dynamical system provided the Hamiltonian is defined and the initial state is specified. Due to the time reversal invariance, the time evolution of a given initial state at t=0 does not have a time-arrow. These problems have been extensively studied by E. C. G. Sudarshan and collaborators, including myself. Here the survival probability, which is the square of the survival amplitude, is an even function of t. Using the survival amplitude as a common framework, we discuss Zeno effects in several processes: the decay of an unstable particle, pion productions in proton-nucleus collisions and the Aharonov-Vardi effect which is another manifestation of the Zeno effect.

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Does the Special Theory of Relativity (STR) forbid the existence of particles traveling with speed greater than the speed of light in vacuo (Tachyons)? Prof. Sudarshan and collaborators O.M. Bilaniuk and V.K. Despande examined this question in 1962 and concluded that STR does not rule out such objects. Now, the momentum 4-vector of a Tachyon is necessarily space-like and consequently, the sign of energy no longer Lorentz-invariant. Thus a Tachyon will be found to have negative energy in certain inertial frames. The authors noted that in the latter, the sign of time-intervals will also be reversed. A negative energy Tachyon traveling backward in time could now be reinterpreted as a positive energy particle traveling forward in time. This reinterpretation could be done consistently as was shown by several examples. A quantum field theory of free spinless Tachyons was suggested by Feinberg (1967). M. Arons and Sudarshan reexamined this model and showed that the model did not possess invariance under the Poincaré group. An alternative version was constructed by them that possessed the desired invariance property but no local commutativity. Subsequently, Dhar and Sudarshan constructed a model of a neutral scalar Tachyon with Yukawa coupling to Fermions. The model had unusual features such as the emission of a Tachyon by a Fermion as a real process and the need for an additional non-local interaction. Prof. Sudarshan, in association with J. Narlikar, studied Tachyons in the context of cosmology in 1976. One of their conclusions is that any primordial Tachyons that might have been created at the beginning are unlikely to have survived to the present era.

Opening remarks at the Open Systems Session (the seventh quest).

The relations between dynamical maps and quantum states of bipartite systems are analyzed from the perspective of quantum conditional probability. In particular, we explore new interesting relations between completely positive maps, which correspond to quantum channels, and states of bipartite systems which correspond to correlations between the initial and final states. The new connection emerges in a natural way from the generalisation of the classical concept of conditional probability.

We develop applications of these relations which prove to be very useful in both directions, either for the classification of positive maps which are not completely positive, the classification of non-decomposable dynamical maps or for the classification of positive partial transpose and entangled states.

The quantum jump approach (QJA), a discretized version of continuous observations of systems coupled to a bath, such as photons, is outlined and it is shown how it avoids the quantum Zeno effect by temporal coarse-graining. Applications to quantum counting processes and spin-boson models for detecting a particle are given.

I briefly review the "decohering histories" or "consistent histories" formulation of quantum theory, due to Griffiths, Omnès, and Gell-Mann and Hartle (and the subject of my graduate work with George Sudarshan). I also sift through the many meanings that have been attached to decohering histories, with an emphasis on the most basic one: Decoherence of appropriate histories is needed to establish that quantum mechanics has the correct classical limit. Then I will describe efforts to find physical mechanisms that do this. Since most work has focused on density matrix versions of decoherence, I'll consider the relation between the two formulations, which historically has not been straightforward. Finally, I'll suggest a line of research that would use recent results by Sudarshan to illuminate this aspect of the classical limit of quantum theory.

It is found that the inclusion of higher derivative terms in the gravitational action along with concepts of phase transition and spontaneous symmetry breaking leads to some novel consequence. The Ricci scalar plays the dual role, like a physical field as well as a geometrical field. One gets Klein-Gordon equation for the emerging field and the corresponding quanta of geometry are called Riccions. For the early universe the model removes singularity along with inflation. In higher dimensional gravity the Riccions can break into spin half particle and antiparticle along with breaking of left-right symmetry. Most tantalizing consequences is the emergence of the physical universe from the geometry in the extreme past. Riccions can Bose condense and may account for the dark matter.

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Talk presented to celebrate Sudarshan's 75th birthday.

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These two days of discussions on physics with special attention to my scientific quests gives me great gratification. So many of my friends who have distinguished themselves as physicists have put their minds to present such excellent expositions. The contributions that I have made over the last half century, when handled by these experts, show their interconnection through the paths chosen by them, and also seem to show the interconnection between these ideas.

Some of the many messages sent to George Sudarshan by those who could not attend this symposium for various reasons.