Dynamical symmetry breaking and the Nambu-Goldstone theorem in the Gaussian wave functional approximation

We analyze the group-theoretical ramifications of the Nambu-Goldstone [NG] theorem in the self-consistent relativistic variational Gaussian wave functional approximation to spinless field theories. In an illustrative example we show how the Nambu-Goldstone theorem would work in the O(N) symmetric $\phi^4$ scalar field theory, if the residual symmetry of the vacuum were lesser than O(N-1), e.g. if the vacuum were O(N-2), or O(N-3),... symmetric. [This does not imply that any of the "lesser" vacua is actually the absolute energy minimum: stability analysis has not been done.] The requisite number of NG bosons would be (2N - 3), or (3N - 6), ... respectively, which may exceed N, the number of elementary fields in the Lagrangian. We show how the requisite new NG bosons would appear even in channels that do not carry the same quantum numbers as one of N "elementary particles" (scalar field quanta, or Castillejo-Dalitz-Dyson [CDD] poles) in the Lagrangian, i.e. in those "flavour" channels that have no CDD poles. The corresponding Nambu-Goldstone bosons are composites (bound states) of pairs of massive elementary (CDD) scalar fields excitations. As a nontrivial example of this method we apply it to the physically more interesting 't Hooft $\sigma$ model (an extended $N_{f} = 2$ bosonic linear $\sigma$ model with four scalar and four pseudoscalar fields), with spontaneously and explicitly broken chiral $O(4) \times O(2) \simeq SU_{\rm R} (2) \times SU_{\rm L}(2) \times U_{\rm A}(1)$ symmetry.Comment: 17 pages, no figure

Discriminator varieties and symbolic computation

AbstractWe look at two aspects of discriminator varieties which could be of considerable interest in symbolic computation:1.discriminator varieties are unitary (i.e., there is always a most general unifier of two unifiable terms), and2.every mathematical problem can be routinely cast in the form†p1 ≈ q1, …, pk ≈ qk implies the equation x ≈ y.Item (l) offers possibilities for implementations in computational logic, and (2) shows that Birkhoff's five rules of inference for equational logic are all one needs to prove theorems in mathematics

Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe

In three spacetime dimensions, general relativity drastically simplifies, becoming a ``topological'' theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2+1)-dimensional vacuum gravity in the setting of a spatially closed universe.Comment: 61 pages, draft of review for Living Reviews; comments, criticisms, additions, missing references welcome; v2: minor changes, added reference

Satisfiability in multi-valued circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.Comment: 50 page

Nature’s Optics and Our Understanding of Light

Optical phenomena visible to everyone abundantly illustrate important ideas in science and mathematics. The phenomena considered include rainbows, sparkling reflections on water, green flashes, earthlight on the moon, glories, daylight, crystals, and the squint moon. The concepts include refraction, wave interference, numerical experiments, asymptotics, Regge poles, polarisation singularities, conical intersections, and visual illusions