First-class constraint: Difference between revisions

Revision as of 23:31, 29 May 2018

Revision as of 21:59, 14 November 2018

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The action {{mvar|S}} is given by

:<math>S[\boldmathbf{A},\sigma]=\int d^dx \frac{1}{4g^2}\eta((\boldmathbf{g}^{-1}\otimes \boldmathbf{g}^{-1})(\boldmathbf{F},\boldmathbf{F}))+\frac{1}{2}\alpha(\boldmathbf{g}^{-1}(D\sigma,D\sigma))</math>

where '''g''' is the Minkowski metric, '''F''' is the [[curvature form]]

:<math>d\boldmathbf{A}+\boldmathbf{A}\wedge\boldmathbf{A}</math>

(no {{mvar|i}}s or {{mvar|g}}s!) where the second term is a formal shorthand for pretending the Lie bracket is a commutator, {{mvar|D}} is the covariant derivative

:D&sigma; = d&sigma; &minus; '''A'''[&sigma;]

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:<math>\rho':\bar{V}\otimes V\rightarrow L</math>

( {{mvar|L}} is self-dual via {{mvar|η}}). The Hamiltonian,

:<math>H_f=\int d^{d-1}x \frac{1}{2}\alpha^{-1}(\pi_\sigma,\pi_\sigma)+\frac{1}{2}\alpha(\vec{D}\sigma\cdot\vec{D}\sigma)-\frac{g^2}{2}\eta(\vec{\pi}_A,\vec{\pi}_A)-\frac{1}{2g^2}\eta(\boldmathbf{B}\cdot \boldmathbf{B})-\eta(\pi_\phi,f)-<\pi_\sigma,\phi[\sigma]>-\eta(\phi,\vec{D}\cdot\vec{\pi}_A).</math>

The last two terms are a linear combination of the Gaussian constraints and we have a whole family of (gauge equivalent)Hamiltonians parametrized by {{mvar|f}}. In fact, since the last three terms vanish for the constrained states, we may drop them.