Joel Feldman

 
Prospective Graduate Students / Postdocs

This faculty member is currently not actively recruiting graduate students or Postdoctoral Fellows, but might consider co-supervision together with another faculty member.

Professor

Research Classification

Mathematical Analysis

Research Interests

Constructive Quantum Field Theory

Relevant Degree Programs

 

Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
Functional integral representations for quantum many-particle systems (2015)

Formal functional integrals are commonly used as theoretical tools and as sources of intuition for predicting phase transitions of many-body systems in Condensed Matter Physics. In this thesis, we derive rigorous versions of these functional integrals for two types of quantum many-particle systems.We begin with a brief review of quantum statistical mechanics in Chapter 2 and the formalism of coherent states in Chapter 3, which form the basis for our analysis in Chapters 4 and 5. In Chapter 4, we study a mixed gas of bosons and/or fermions interacting on a finite lattice, with a general Hamiltonian that preserves the total number of particles in each species. We rigorously derive a functional integral representation for the partition function, which employs a large-field cutoff for the boson fields. We then expand the resulting “action” in powers of the fields and find a recursion relation for the coefficients. In the case of a two-body interaction (such as the Coulomb interaction), we also find bounds on the coefficients, which give a domain of analyticity for the action. This domain is large enough for use of the action in the functional integral, provided that the large-field cutoffs are taken to grow not too quickly. In Chapter 5, we study a system of electrons and phonons interacting in a finite lattice, using the Holstein Hamiltonian. Again, we rigorously derive a coherent-state functional integral representation for the partition function of this system and then prove that the “action” in the functional integral is an entire-analytic function of the fields. However, since the Holstein Hamiltonian does not preserve the total number of bosons, the approach from Chapter 4 requires some modification. In particular, we repeatedly use Duhamel expansions in powers of the interaction, rather than sums over particle numbers.

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Asymptotics for Fermi curves of electric and magnetic periodic fields (2009)

This work is concerned with some geometrical properties of (complex) Fermi curves of electric and magnetic periodic fields. These are analytic curves in C² that arise from the study of the eigenvalue problem for periodic Schroedinger operators. More specifically, we characterize a certain class of these curves in the region of C² where at least one of the coordinates has "large" imaginary part. The new results obtained in this thesis extend previous results in the absence of magnetic field to the case of "small" magnetic field. Our theorems can be used to show that generically these Fermi curves belong to a class of Riemann surfaces of infinite genus.

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