Professor of Physics
Northeastern University
Boston, MA 02115
Tel: (617) 373-2954
E-mail: m.vaughn at neu dot edu

Ph. D., Purdue University, 1960
Field of Specialization: Elementary particle theory

### Mathematical Physics

After many years, I have finished a graduate level textbook and reference work on mathematical physics with the mundane title "Introduction to Mathematical Physics". The important new feature of this book is the introduction of differential forms and other geometrical concepts early on (in chapter 3). There are also introductions to nonlinear problems, and to group theory, and a reduced emphasis on the special functions that arise in the solution of Laplace's equation and its relatives.

"Introduction to Mathematical Physics" Vaughn

an early link will be to the Google books site. The book is also available from the usual online booksellers.

A collection of problem solutions (not complete) for instructors who adopt the book for a course is available from the publisher.

There is a page of errata for the book. Please e-mail me questions, comments, corrections and any other feedback.

### Teaching

In the Fall 2008 semester, I am teaching
• PHYG301 -- Classical Mechanics and Mathematical Physics (first-year physics graduate students)
• PHYG321 -- Computational Physics (second-year physics graduate students)
In the Spring 2008 semester, I taught
• PHYU613 -- Nuclear and Particle Physics (upperclass science majors)
In the Fall 2007 Semester, I taught
• PHYU111 -- Introduction to Astronomy (undergraduate science elective)
• PHYG302 -- Electromagnetic Theory (first-year physics graduate students)

### Research

Beyond completing the mathematical physics book noted above, I have a number of research interests:

#### PT-symmetry in quantum mechanics

About a year ago, Carl Bender gave a marvelous talk at Northeastern describing a number of examples in which ostensibly non-Hermitian operators have purely real spectra. In these examples, it is possible to define a linear operator P (analogous to parity) and an antilinear operator T (analogous to time reversal), such that the Hamiltonian is invariant under the product PT, though not under either of them alone. There are now many people working on trying to find other examples with PT-symmetry, and perhaps finding new physical applications of these models. I will expand this section in due course.

#### gauge dependence of the electron propagator in QED

Some colleagues recently had the clever idea of trying to measure the exponent of the mass-shell singularity of the electron propagator, since it has been known since the mid-1950s that this singularity is not a simple pole. However, it is also well-known that the exponent of the singularity is actually gauge-dependent, so it is not a measurable quantity. I have written a short note to explain this, using dimensional regularization and mass-shell subtraction in a general $R_\xi$ gauge (the textbook calculations I have seen of the propagator that use dimensional regularization are in Feynman gauge).

This note will not be published, since the result is not new. However, it may have some pedagogical value to illustrate the power of dimensional regularization to eliminate possible infrared singularities that sometimes appear when introducing a small photon mass and later taking it to zero.

#### higher symmetries in condensed matter systems

In the late 1990s, I worked with Bob Markiewicz in studying the role of higher symmetries in condensed matter physics, especially in the theory of the cuprate high $T_c$ superconductors and other quasi-two-dimensional condensed matter systems.

#### renormalization group and unified gauge theories

This section is under construction.

#### Selected Publications

• Michael T. Vaughn, Introduction to Mathematical Physics, Wiley-VCH (2007).

• M.E. Machacek and M.T. Vaughn, "Two-Loop Renormalization Group Equations in a General Quantum Field Theory I. Wave Function Renormalization", Nucl. Phys. B222, 83 (1983); "... II. Yukawa Couplings", ibid. B236, 221 (1984); "... III. Scalar Quartic Couplings", ibid. B249, 83 (1985).

• R.S. Markiewicz and M.T. Vaughn, "Stripes, pseudogaps, and SO(6) in the cuprate superconductors", J. Phys. Chem. Sol. 59, 1737 (1998) (cond-mat/9709137).
• R.S. Markiewicz and M.T. Vaughn, "Classification of the Van Hove Scenario as an SO(8) Spectrum Generating Algebra", Phys. Rev. B57, 14052 (1998) (cond-mat/9802078).
• R.S. Markiewicz and M.T. Vaughn, "Higher Symmetries in Condensed Matter Physics", in "Particles, Strings and Cosmology -- PASCOS98", P. Nath (ed.) , World Scientific (Singapore, 1999) (cond-mat/9809119).

This is a work under construction. If you are missing something, please send me e-mail.
last update 8 September 2008