arXiv subjects
Publications and source records attributed to Donald Spector.
What can we hope to learn about brains from the free energy principle? In adopting the "primordial soup" physical model, Bruineberg et al. perpetuate the unsupported notion that the free energy principle has a meaningful physical--and neuronal--interpretation. We examine how minimization of free energy arises in physical contexts, and what this can and cannot tell us about brains.
We examine the quantization of pseudoclassical dynamical systems, models that have classically anticommuting variables, in the Schr\"odinger picture. We quantize these systems, which can be viewed as classical models of particle spin, using the generalized Gupta-Bleuler method as well as the reduced phase space method in even dimensions. With minimal modifications, the standard constructions of Schr\"odinger quantum mechanics of constrained systems work for pseudoclassical systems. We generalize the standard Schr\"odinger norm and implement the correct adjointness properties of observables and constraints. We construct the state space corresponding to spinors as physical wave functions of anticommuting variables, finding that there are superselection sectors in both the physical and ghost subspaces. The physical states are isomorphic to those of the Dirac-K\"ahler formulation of fermions though the inner product in Dirac-K\"ahler theory is not equivalent to ours.
This paper identifies a new class of shape invariant models. These models are based on extensions of conventional quantum mechanics that satisfy a string-motivated minimal length uncertainty relation. An important feature of our construction is the pairing of operators that are not adjoints of each other. The results in this paper thus show the broader applicability of shape invariance to exactly solvable systems.
We systematically include central charges into supersymmetric quantum mechanics formulated on curved Euclidean spaces, and explain how the background geometry manifests itself on states of the theory. In particular, we show in detail how, from the point of view of non-relativistic d=1 world-line physics, one can infer the existence of target space dualities typically associated with string theory. We also explain in detail how the presence of a non-trivial supersymmetry central charge restricts the background geometry in which a particle may propagate.
We show that shape invariance appears when a quantum mechanical model is invariant under a centrally extended superalgebra endowed with an additional symmetry generator, which we dub the shift operator. The familiar mathematical and physical results of shape invariance then arise from the BPS structure associated with this shift operator. The shift operator also ensures that there is a one-to-one correspondence between the energy levels of such a model and the energies of the BPS-saturating states. These findings thus provide a more comprehensive algebraic setting for understanding shape invariance.
We identify a class of point-particle models that exhibit a target-space duality. This duality arises from a construction based on supersymmetric quantum mechanics with a non-vanishing central charge. Motivated by analogies to string theory, we are led to speculate regarding mechanisms for restricting the background geometry.
I examine quantum mechanical Hamiltonians with partial supersymmetry, and explore two main applications. First, I analyze a theory with a logarithmic spectrum, and show how to use partial supersymmetry to reveal the underlying structure of this theory. This method reveals an intriguing equivalence between two formulations of this theory, one of which is one-dimensional, and the other of which is infinite-dimensional. Second, I demonstrate the use of partial supersymmetry as a tool to obtain the asymptotic energy levels in non-relativistic quantum mechanics in an exceptionally easy way. In the end, I discuss possible extensions of this work, including the possible connections between partial supersymmetry and renormalization group arguments.
We analyze a potential that produces background charges which are automatically quantized. This introduces a new mechanism for charge quantization, although so far it has only been implemented for background charges. We show that this same mechanism can also lead to an alternative means of hiding extra dimensions that is analogous to the Kaluza-Klein approach.
We study some of the algebraic properties of the non-relativistic monopole. We find that we can construct theories that possess an exotic conserved fermionic charge that squares to the Casimir of the rotation group, yet do not possess an ordinary supersymmetry. This is in contrast to previous known examples with such exotic fermionic charges. We proceed to show that the presence of the exotic fermionic charge in the non-supersymmetric theory can nonetheless be understood using supersymmetric techniques, providing yet another example of the usefulness of supersymmetry in understanding non-supersymmetric theories.
We show in a model-independent way that, in the background of a topological soliton or instanton that saturates a Bogomol'nyi bound, the fermion and boson excitation spectra of non-zero modes cancel at the one-loop level. This generalizes D'Adda and DiVecchia's result for some specific instanton models. Our method also establishes, again in a model-independent way, the generality of the connection between zero modes in topologically non-trivial backgrounds and index theorems.
We find examples of duality among quantum theories that are related to arithmetic functions by identifying distinct Hamiltonians that have identical partition functions at suitably related coupling constants or temperatures. We are led to this after first developing the notion of partial supersymmetry-in which some, but not all, of the operators of a theory have superpartners-and using it to construct fermionic and parafermionic thermal partition functions, and to derive some number theoretic identities. In the process, we also find a bosonic analogue of the Witten index, and use this, too, to obtain some number theoretic results related to the Riemann zeta function.
Using the theory of supersymmetric anyons, I extend the definition of the Witten index to 2+1 dimensions so as to accommodate the existence of anyon spin and statistics. I then demonstrate that, although in general the index receives irrational and complex contributions from anyonic states, the overall index is always integral, and I consider some of the implications and interpretations of this result.
We show that the Calogero and Calogero-Sutherland models possess an N-body generalization of shape invariance. We obtain the operator representation that gives rise to this result, and discuss the implications of this result, including the possibility of solving these models using algebraic methods based on this shape invariance. Our representation gives us a natural way to construct supersymmetric generalizations of these models, which are interesting both in their own right and for the insights they offer in connection with the exact solubility of these models.
I give an interpretation of the fundamental theorem of algebra based on supersymmetry and the Witten index. The argument gives a physical explanation of why a real polynomial of degree $n$ need not have $n$ real zeroes, while a complex polynomial of degree $n$ must have $n$ complex zeroes. This paper also addresses in a general and model-independent way the statistics of the perturbative ground states (the states which correspond to classical vacua) in supersymmetric theories with complex and with real superfields.
We give model-independent arguments, valid in nearly any number of spacetime dimensions, that topological solitons and instantons satisfy Bogomol'nyi-type bounds and, when these bounds are saturated, satisfy self-duality equations. In the supersymmetric case, we also show that, in spacetime dimensions greater than two, theories with topological charges necessarily exhibit extended supersymmetry, in which the topological charge appears as the central charge. The significance of our arguments lies in their generality. In the supersymmetric case, we obtain insight into the contrast observed between topological charges in 1+1 and higher dimensional models. The centerpiece of our method is to require that the supersymmetric extension of a generic (non-supersymmetric) field theory be self-consistent. Our discussion of supersymmetric extensions is quite detailed, and introduces the notion of the "associated superfield" to construct such extensions.
This talk summarizes our recent work establishing an algebraic, model-independent basis for the existence of \B bounds and \B equations for topologically non-trivial solitons and instantons. Our arguments use supersymmetry in an essential way to understand both supersymmetric and non-supersymmetric theories. Our arguments are constructive and work in nearly any number of dimensions. Presented at and to appear in the proceedings of the XX^{th} International Colloquium on Group Theoretical Methods in Physics, Osaka, July 1994.
I provide an extremely simple argument that the kink-type solitons in certain theories are fermionic. The argument is based on the Witten index, but can in fact be used to determine soliton statistics in non-supersymmetric theories as well.
We use separation of variables as a tool to identify and to analyze exactly soluble time-dependent quantum mechanical potentials. By considering the most general possible time-dependent re-definition of the spatial coordinate, as well as general transformations on the wavefunctions, we show that separation of variables applies and exact solubility occurs only in a very restricted class of time-dependent models. We consider the formal structure underlying our findings, and the relationship between our results and other work on time-dependent potentials. As an application of our methods, we apply our results to the calculations of propagators.