Physics | Quantum Many-Body Theory II - Nuclear
P626 | 3331 | Serot

P626: NUCLEAR MANY-BODY PHYSICS
Brian D. Serot
Spring 2001
MWF 10:10--11:00 a.m.

Text: Theoretical Nuclear and Subnuclear Physics, by John Dirk Walecka,
Oxford University Press, 1995.

Motivation: To present the modern theoretical basis for the study
of the nuclear many-body problem and to show how this theory based
quantum chromodynamics (QCD).

Since we won't be able to cover all of the topics in detail, we will
try to exhibit the most basic and important results.
fill in the omissions.

GOALS and PRELIMINARY OUTLINE [estimated weeks]:

1. To learn some basic nuclear phenomenology [3]:
A. The characteristics of the NN interaction,
B. The properties of nuclear matter; summary of nonrelativistic
Brueckner theory,
C. The bulk (semi-empirical mass formula) and single-particle
(shell-model) properties of medium to heavy nuclei,
D. (?) The Skyrme potential; nuclear matter calculations,
E. (?) The structure of light nuclei (deuteron, triton, helion,
alpha) and the necessity for NNN forces.

2. To learn some necessary elements of relativistic quantum field
theory [2]:
A. The Dirac equation; second quantization (?),
B. Electromagnetic gauge invariance and the QED lagrangian,
C. Local color SU(3) symmetry and the QCD lagrangian:
i. Asymptotic freedom; use perturbative QCD here,
ii. Confinement: why all observables and observed particles
iii. Why you want to use QHD for nuclear physics; we'll start
with nucleons and pions,
D. The basics of effective field theory (EFT):
i. Separation of scales (ideas of Weinberg, Georgi):
"known long-range interactions constrained by symmetries
and a complete set of generic short-range interactions,"
ii. Matching (where it works, e.g, weak interactions and
Fermi's theory, and where it fails, e.g, nuclear
physics; why we must fit parameters in QHD rather than
match them),
iii. How symmetries provide constraints.

3. To learn about chiral symmetry and spontaneous symmetry breaking [2]:
A. Discrete symmetries (P, C, T),
B. Continuous global symmetries [i.e., SU(2)_L \times SU(2)_R],
C. Spontaneoulsy broken (chiral) symmetry and Goldstone
bosons (pions),
D. The linear Sigma model,
E. The nonlinear Sigma model,
F. Chiral perturbation theory:
i. Overall strategy (see Donoghue et al.),
ii. Expansion parameter(s),
iii. Explicit forms of pionic lagrangians,
iv. Brief discussion of phenomenology and status,
v. Vector meson "saturation" and "dominance".
G. Naive dimensional analysis (NDA),

4. To learn about QHD and how to construct the EFT lagrangian [4]:
A. Pions and nucleons; how this fits with chiral perturbation theory,
B. Symmetry constraints: Lorentz covariance, P, C, T, isospin,
chiral invariance,
C. What observables we will be primarily interested in
(for now); more justification later,
D. Incorporation of collective degrees of freedom:
i. Vector mesons: vector-meson dominance (VMD); relation to
one-boson-exchange potentials (OBEP),
ii. Delta baryon: Chew--Low theory; why one uses the Delta; why
we won't need it now,
iii. Scalar meson: Correlated two-pion exchange; relation to
OBEP; relation to phonons in CM physics; why we
*really* need this field,
iv. Don't impose unnecessary constraints on couplings (as in
derivations based on linear sigma models).
i. Introduction of EM interactions (no details),
ii. Inclusion of nucleonic EM structure through a derivative
expansion,
iii. Inclusion of nucleonic QCD structure through meson
nonlinearities, etc.,
F. Construction of the most general VMD lagrangian:
i. NDA power counting, naturalness,
ii. Truncation (to be verified a posteriori),
iii. Redundancy (technical and practical),
G. Other choices of field variables (Point Couplings; Skyrme Pot'l).

5. To learn the basic ideas of density functional theory (DFT) [1]:
A. Hohenberg--Kohn theorem; thermodynamic analogy,
B. Kohn--Sham theorem (Kohn's 1999 proof); justifies choice of
observables,
C. "Hartree Dominance" and why mean-field theory should be
a good approximation (to be verified a posteriori).

6. To learn about relativistic mean-field theory (MFT) [1]:
A. Mean-field energy functional,
B. Dirac--Hartree equations,
C. Results of empirical fits; justification of truncation and
naturalness,
D. Other coordinate choices (examples),
E. Determination of model parameters (how many?),
F. (?) Justification of large potentials: QCD energy scales;

7. To learn what to do in the future [1]:
A. Isovector observables and response: pions and Deltas,
B. Explicit many-body corrections: separation of short-range
(vacuum) and long-range effects; persistence of
naturalness (Y. Hu thesis),
C. Excited states and "conserving" random-phase approximation,
D. Renormalization group methods (dilute Fermi gas??),
E. Connection to underlying QCD: a well-defined procedure exists
(in principle); all we have to do is match the QHD
parameters to QCD! (We don't have to calculate with QCD!)

8. Summary and Outlook