169,286 research outputs found
Zariski Quantization as Second Quantization
The Zariski quantization is one of the strong candidates for a quantization
of the Nambu-Poisson bracket. In this paper, we apply the Zariski quantization
for first quantized field theories, such as superstring and supermembrane
theories, and clarify physical meaning of the Zariski quantization. The first
quantized field theories need not to possess the Nambu-Poisson structure.
First, we construct a natural metric for the spaces on which Zariski product
acts in order to apply the Zariski quantization for field theories. This metric
is invariant under a gauge transformation generated by the Zariski quantized
Nambu-Poisson bracket. Second, we perform the Zariski quantization of
superstring and supermembrane theories as examples. We find flat directions,
which indicate that the Zariski quantized theories describe many-body systems.
We also find that pair creations and annihilations occur among the many bodies
introduced by the Zariski quantization, by studying a simple model. These facts
imply that the Zariski quantization is a second quantization. Moreover, the
Zariski quantization preserves supersymmetries of the first quantized field
theories. Thus, we can obtain second quantized theories of superstring and
supermembranes by performing the Zariski quantization of the superstring and
supermembrane theories.Comment: 18 pages, 2 figure
Inclusions of second quantization algebras
In this note we study inclusions of second quantization algebras, namely
inclusions of von Neumann algebras on the Fock space of a separable complex
Hilbert space H, generated by the Weyl unitaries with test functions in closed,
real linear subspaces of H. We show that the class of irreducible inclusions of
standard second quantization algebras is non empty, and that they are depth two
inclusions, namely the third relative commutant of the Jones' tower is a
factor. When the smaller vector space has codimension n into the bigger, we
prove that the corresponding inclusion of second quantization algebras is given
by a cross product with R^n. This shows in particular that the inlcusions
studied in hep-th/9703129, namely the inclusion of the observable algebra
corresponding to a bounded interval for the (n+p)-th derivative of the current
algebra on the real line into the observable algebra for the same interval and
the n-th derivative theory is given by a cross product with R^p. On the
contrary, when the codimension is infinite, we show that the inclusion may be
non regular (cf. M. Enock, R. Nest, J. Funct. Anal. 137 (1996), 466-543), hence
do not correspond to a cross product with a locally compact group.Comment: LaTex, 9 pages, requires cmsams-l.cl
Bosonization method for second super quantization
A bosonic-fermionic correspondence allows an analytic definition of
functional super derivative, in particular, and a bosonic functional calculus,
in general, on Bargmann- Gelfand triples for the second super quantization. A
Feynman integral for the super transformation matrix elements in terms of
bosonic anti-normal Berezin symbols is rigorously constructed.Comment: In memoriam of F. A. Berezin, accepted in Journal of Nonlinear
Mathematical Physics, 15 page
Second Quantization and the Spectral Action
We consider both the bosonic and fermionic second quantization of spectral
triples in the presence of a chemical potential. We show that the von Neumann
entropy and the average energy of the Gibbs state defined by the bosonic and
fermionic grand partition function can be expressed as spectral actions. It
turns out that all spectral action coefficients can be given in terms of the
modified Bessel functions. In the fermionic case, we show that the spectral
coefficients for the von Neumann entropy, in the limit when the chemical
potential approaches can be expressed in terms of the Riemann zeta
function. This recovers a result of Chamseddine-Connes-van Suijlekom.Comment: Author list is expanded. The calculations in the new version are
extended to two more Hamiltonians. New references adde
Second Quantization of the Wilson Loop
Treating the QCD Wilson loop as amplitude for the propagation of the first
quantized particle we develop the second quantization of the same propagation.
The operator of the particle position (the endpoint of the
"open string") is introduced as a limit of the large Hermitean matrix. We
then derive the set of equations for the expectation values of the vertex
operators \VEV{ V(k_1)\dots V(k_n)} . The remarkable property of these
equations is that they can be expanded at small momenta (less than the QCD mass
scale), and solved for expansion coefficients. This provides the relations for
multiple commutators of position operator, which can be used to construct this
operator. We employ the noncommutative probability theory and find the
expansion of the operator in terms of products of creation
operators . In general, there are some free parameters left
in this expansion. In two dimensions we fix parameters uniquely from the
symplectic invariance. The Fock space of our theory is much smaller than that
of perturbative QCD, where the creation and annihilation operators were
labelled by continuous momenta. In our case this is a space generated by creation operators. The corresponding states are given by all sentences made
of the four letter words. We discuss the implication of this construction for
the mass spectra of mesons and glueballs.Comment: 41 pages, latex, 3 figures and 3 Mathematica files uuencode
Entanglement in The Second Quantization Formalism
We study properties of entangled systems in the (mainly non-relativistic)
second quantization formalism. This is then applied to interacting and
non-interacting bosons and fermions and the differences between the two are
discussed. We present a general formalism to show how entanglement changes with
the change of modes of the system. This is illustrated with examples such as
the Bose condensation and the Unruh effect. It is then shown that a
non-interacting collection of fermions at zero temperature can be entangled in
spin providing that their distances do not exceed the inverse Fermi wavenumber.
Beyond this distance all bipartite entanglement vanishes, although classical
correlations still persist. We compute the entanglement of formation as well as
the mutual information for two spin-correlated electrons as a function of their
distance. The analogous non-interacting collection of bosons displays no
entanglement in the internal degrees of freedom. We show how to generalize our
analysis of the entanglement in the internal degrees of freedom to an arbitrary
number of particles.Comment: 11 pages, no figures, a few typos corrected and some references adde
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