Particles, Fields and Topology

The following sections are included:

• Preface
• Contents

Hegel wrote: “The owl of Minerva spreads its wings only with the falling of the dusk.”

To this I may add Edward Hallett Carr, in What Is History?: “What is history?, our answer, consciously or unconsciously, reflects our own position in time, and forms part of our answer to the broader question, what view we take of the society in which we live”…

The dynamics of field theories in domains with boundaries is governed not only by field equations but also by its boundary conditions. In this chapter, we analyze the most general energy preserving boundary conditions for scalar field theories and the global properties of the space defined by all of them. In general, it has two connected components. In the case of complex scalar fields, we show the compatibility of these boundary conditions with the ones that preserve electric charge.

This contributory chapter begins with our fond and sincere reminiscences about our beloved Prof. A.P. Balachandran. In the main part, we discuss our recent formulation of quantum mechanics on (1+1)D noncommutative space-time using Hilbert–Schmidt operators. As an application, we demonstrate how geometrical phase in a system of time-dependent forced harmonic oscillator living in the Moyal space-time can emerge.

We construct the fuzzy spaces based on the three non-trivial coadjoint orbits of the exceptional simple Lie group, G₂.

This short contribution is a review of some of the scientific results that were achieved in Syracuse during my stay as a PhD student under the supervision of A.P. Balachandran from 1990 to 1994. It was an exciting period during which we worked closely with other PhD students and many visiting researchers on the mathematical structures of quantum mechanics. The research focused on some fundamental issues that the physical community managed to understand but put in a full coherent theory only much later.

It is shown that a mass scalar field probe can be used to reveal the near-horizon conformal structure of a large class of black holes. The logarithmic correction to black hole entropy follows from the algebraic properties of the near-horizon Klein–Gordon operator. The black hole decay can be analyzed within this formalism as a geodesic motion in an appropriate moduli space.

The recent groupoidal description of Schwinger’s picture of quantum mechanics is used to provide a new approach to the notion of quantum field. Starting with Feynman’s quantum mechanics, where classical paths play a key role, a groupoidal interpretation of them is provided by reckoning the basic notions involved on them, most singularly the notion of clocks. It is found that natural requisites about the use of clocks in describing quantum systems lead naturally to the notion of histories as a class of homomorphisms of groupoids, and such notion leads directly to Feynman’s notion of classical paths. Similar arguments can be used when replacing clocks by reference frames. There the notion of universal histories of quantum systems and an associated notion of field emerge naturally. Moreover, the analysis of the properties of clocks in quantum systems leads to a characterization of semidirect products of groupoids.

It is an honour and a pleasure to be able to contribute to this Festschrift dedicated to Bal, but I want to write on a subject he does not like, and does not think much of, hoping to convince him that it is important and interesting, in spite of the fact that the work is not even finished: galaxies, which are sort of half way between elementary particles and cosmology but important for both. Galaxies are huge: for a typical one SGC1560 light takes ∼25000 years to go from one end to the other; it has a visible mass estimated to ∼8. · 10⁸M_⊙; it is more like a pancake than a sphere with an axis ratio ∼5 : 1; it rotates, with an edge velocity ∼36 km/s, the focus of our attention. Outside its visible part, there is a lot of ionized gas circling it. The famous astronomer Vera Rubin discovered that the velocity of this gas does not fall down with distance as one might naively expect from Kepler law but stays roughly constant. Now, after 400 years arguing with Kepler’s law, and with Newton’s law that explains it, it is sort of sacrilegious, but what else can one do? Astrophysicists are resourceful: they invented ‘dark matter’. What you see, ‘baryonic matter’, is only a small percentage (5%) of the total mass; the rest is invisible and only interacts through gravitation, and it is distributed over a much larger volume that the visible mass. That explains the constant gas velocities. Humans are ignorant and pretentious, yet having to admit that we know nothing about what 95% of the Universe is really made of is a bit too much. Other physicists, mostly not astro-, are skeptical and try to find alternative explanations. I joined the “no dark matter” club, together with my colleague and long time friend Yogendra Srivastava…

Chaotic dynamics of the mass deformed ABJM model is explored. To do so, we consider spatially uniform fields and obtain a family of reduced effective Lagrangians by tracing over ansatz configurations involving fuzzy two-spheres with collective time dependence. We examine how the largest Lyapunov exponent, λ_L, changes as a function of E/N², where N is the matrix size. In particular, we inspect the temperature dependence of λ_L and present upper bounds on the temperature above which λ_L values comply with the MSS bound, λ_L ≤ 2πT , and below which it will eventually be not obeyed.

We describe how to associate a Hopf algebroid to a noncommutative principal bundle. This is a quantization of the gauge groupoid of a classical bundle. The gauge group of the noncommutative bundle is isomorphic to the group of bisections of the algebroid. As examples, we give a monopole bundle over a quantum sphere and noncommutative bundles over a point associated with Taft algebras.

Let us begin with a celebrated anecdote involving an Indian scientist from Tamil Nadu: Ramanujan. The great number theorist G. Hardy recalls: I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.” It is not clear what would Ramanujan had answered if Hardy had taken cab 316. The Penguin Dictionary of Curious and Interesting Numbers jumps from 306 to 319 (which cannot be represented as the sum of fewer than 19 4th powers). Even a perusal of Wikipedia does not illuminate; it is a centered triangular number and a centered heptagonal number, a psalm of John (3:16), the area code of Wichita (Texas) and the year of the consulship of Sabinus and Rufinus…

Edge modes in gauge theories, whose raison d’être is in the nature of the test functions used for imposing the Gauss law, have implications in many physical contexts. I discuss two such cases: (1) how edge modes are related to the interface term in the BFK formula and how they generate the so-called contact term for entanglement entropy in gauge theories and (2) how they describe the dynamics of particles in generalizing the Einstein–Infeld–Hoffmann approach to particle dynamics in theories of gravity.

Loop braid groups characterize the exchange of extended objects, namely loops, in three-dimensional space generalizing the notion of braid groups that describe the exchange of point particles in two-dimensional space. Their interest in physics stems from the fact that they capture anyonic statistics in three dimensions which is otherwise known to only exist for point particles on the plane. Here, we explore another direction where the algebraic relations of the loop braid groups can play a role: quantum integrable models. We show that the symmetric loop braid group can naturally give rise to solutions of the Yang–Baxter equation, proving the integrability of certain models through the RTT relation. For certain representations of the symmetric loop braid group, we obtain integrable deformations of the XXX-, XXZ- and XY Z-spin chains.

This chapter presents in a self-contained way A. Uhlmann’s celebrated theorem of monotonicity of the relative entropy under completely positive and trace preserving maps. The theorem is presented in its more general form and meaningful examples are given.

We study the exact solutions of both, massless and massive, scalar field theories on the noncommutative AdS₂. We also discuss some important limits in order to compare with known results.

The light cone formalism of a massive scalar field has been shown by Dirac to have many advantages. But it is not manifestly Lorentz invariant. We will show that this is a feature not a bug: Lorentz invariance is indeed a symmetry but in a different sense as defined by Drinfel’d. The key idea is that the mass shell (mass hyperboloid) is a Poisson–Lie group: there is a non-abelian group multiplication and non-zero Poisson brackets between components of four-momentum. Rotations form the dual group of the hyperboloid in the sense of Drinfel’d. Infinitesimal Lorentz transformations form a Lie bi-algebra.

The following sections are included:

• Syracuse to Trieste
• Asymmetric Texture
• Family Symmetry
• Acknowledgements
• References

An algebraic approach to the study of entanglement entropy of quantum systems is reviewed. Starting with a state on a C*-algebra, one can construct a density operator describing the state in the GNS representation space. Applications of this approach to the study of entanglement measures for systems of identical particles are outlined. The ambiguities in the definition of entropy within this approach are then related to the action of unitaries in the commutant of the representation and their relation to modular theory explained.

Earlier work of Balachandran and friends provided a map from algebras to field theories. These methods provide insight into quantum gauge theories and anomalies. In this chapter, we take the reader from the coadjoint representation of the Virasoro algebra to four- (and higher-) dimensional gravitation and cosmology. The protagonist in this story is a component of the projective connection, the diffeomorphism field, which straddles between the one-dimensional world of initial data in string theories to cosmology in four dimensions. We review mathematical intuition that ties projective geometry to the Virasoro algebra, the Thomas–Whitehead (TW) gravitational action that gives the diffeomorphism field dynamics and the building blocks for gauge projective Dirac action.

The conventional Rosenfeld–Bergmann–Dirac constrained Hamiltonian algorithm applied to Einstein–Yang–Mills theory is shown to be equivalent to a local gauge theoretic extension of Cartan’s invariant integral approach to classical mechanics. In addition, the Hamiltonian generators of Legendre-projectable space-time diffeomorphism and gauge symmetries are derived directly as vanishing Noether charges. This leads directly to their interpretation as delivering the correct symmetry variations of both configuration and momentum field variables.

The renormalization group is one of the foundational concepts in quantum field theory. We discuss the role played by the renormalization group in string theory. One concrete example is described that involves an exact form of the renormalization group equation written down by Wilson and is commonly known as Exact Renormalization Group (ERG). This example is in the context of AdS/CFT correspondence in string theory. Starting from the ERG equation for the boundary conformal field theory, it is shown that one can obtain what has come to be called the holographic renormalization group equation. This procedure can lead to a derivation of the AdS/CFT correspondence. Another application of the ERG formalism is the derivation of gauge invariant and background invariant equations of motion for the fields of open and closed strings. Finally, a speculation that some version of the renormalization group is the underlying symmetry principle of string theory is also outlined.

Given a vector space V which is the tensor product of vector spaces A and B, we reconstruct A and B from the family of simple tensors a ⊗ b within V . In an application to quantum mechanics, one would be reconstructing the component subsystems of a composite system from its unentangled pure states. Our constructions can be viewed as instances of the category-theoretic concepts of functor and natural isomorphism, and we use this to bring out the intuition behind these concepts and also to critique them. Also presented are some suggestions for further work, including a hoped-for application to entanglement entropy in quantum field theory.

Using the exact solutions to the field equation for a massive scalar field on noncommutative AdS₂, we apply the AdS/CFT correspondence principle to obtain an exact result for the associated two-point function on the conformal boundary. The answer satisfies conformal invariance and has the correct commutative limit and massless limit.

We reconsider the representation of the quantum transition amplitude as a sum over continuous (in time) histories. Continuous histories may be decomposed into three sets: those histories corresponding to paths in configuration space which are globally (or piece-wise) differentiable with respect to time and the remaining paths which are not differentiable. Under reasonable assumptions, we show that the contribution from the first two sets of paths can be expressed in terms of unitary, anti-Zeno propagators. We can then show that both the transition amplitude and probability are expressible as the sum of holonomies of the canonical one-form on the set of differentiable paths in phase space connecting the initial and final states.

The geometry at the infinitesimal level cannot be described by the conventional axioms of geometry was the famous quote of Riemann himself. He went ahead to suggest that the physics of measuring process of length will affect it. Classical general relativity relies heavily on Riemannian geometry. We argue that the noncommutative geometry naturally fits this scheme at extremely dense astrophysical systems. In the last two decades, several over-luminous type Ia supernovae have been observed, which predict possible white dwarfs violating Chandrasekhar limit significantly. We provide a model using squashed fuzzy sphere to describe such systems by allowing extra mass to be accumulated.

The SU(N) Yang–Mills matrix model admits self-dual and anti-self-dual instantons. The Euclidean Dirac equation in an instanton background has n₊ positive and n₋ negative chirality zero modes. We show that the index (n₊ − n₋) is equal to a suitably defined instanton charge. Further, we show that the path integral measure is not invariant under a chiral rotation and relate the non-invariance of the measure to the index of the Dirac operator. Axial symmetry is broken anomalously to a finite group. For N_f fundamental fermions, this residual symmetry is ${\mathbb{Z}}_{2N_f}$, whereas for adjoint quarks, it is ${\mathbb{Z}}_{4N_f}$.

The correspondence between quantum mechanics and noncommutative geometry is illustrated in the context of the noncommutative ${\text{AdS}}_{(\theta )}^{(2)}/{\text{CFT}}_1$ duality where CFT₁ is identified as conformal quantum mechanics. This model is conjectured here to describe the gauge/gravity correspondence in one dimension.