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Início
Agenda
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- 30 de outubro de 2019
16h30
Sala
F-149 -
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Paulo
A. Faria da Veiga
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ICMC-USP
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On
Thermodynamic and Ultraviolet Stability of Yang-Mills
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We prove thermodynamic and ultraviolet stable
stability bounds for the pure Yang-Mills relativistic quantum
theory in an imaginary-time, functional integral formulation. We
consider the gauge groups G=U(N), SU(N) and let d(N) denote
their Lie algebra dimensions. We start with a finite hypercubic
lattice \Lambda\subset aZ^d$, d=2,3,4$, a\in(0,1], L\in\mathbb N
sites on a side, and with free boundary conditions. The Wilson
partition function Z_{\Lambda,a}\equiv Z_{\Lambda,a,g^2,d} is
used, where the action is a sum over gauge-invariant plaquette
actions with a pre-factor (a^{d-4}/g^2), where g^2\in(0,g_0^2],
0<g_0<\infty, defines the gauge coupling. Each plaquette
action is pointwise positive. Formally, in the continuum limit
a\searrow 0, this action gives the well-known Yang-Mills action.
Either by using the positivity property and neglecting some of
the plaquette actions or by fixing an enhanced temporal gauge,
which involves gauging away the bond variables belonging to a
maximal tree in \Lambda, and which does not alter the value of
Z_{\Lambda,a}, we retain only \Lambda_r bond variables.
\Lambda_r is of order [(d-1)L^d], for large L.
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We prove that the normalized partition function
Z^n_{\Lambda,a}=
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(a^{(d-4)}/g^2)^{d(N)\Lambda_r/2} Z_{\Lambda,a}$
satisfies the stability bounds
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e^{c_\ell d(N)\Lambda_r}\leq Z^n_{\Lambda,a}\leq
e^{c_ud(N)\Lambda_r}, with finite c_\ell,\,c_u\in\mathbb R
independent of L, the lattice spacing 'a' and g^2.
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In other words, we have extracted the exact
singular behavior of the finite lattice free-energy in the
continuum limit 'a' goes to zero. For the normalized free energy
f_{\Lambda,a}^n=[d(N)\,\Lambda_r]^{-1}\,\ln Z^n_{\Lambda,a}, our
stability bounds imply, at least in the sense of subsequences,
that a finite thermodynamic limit
f^n_a\equiv\lim_{\Lambda\nearrow a\mathbb Z^d} f_{\Lambda,a}^n
exists. Subsequently, a subsequential finite continuum limit
f^n\equiv \lim_{a\searrow 0}f^n_a also exists.
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