First-class constraint: Difference between revisions

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==Examples==

LookConsider at the dynamics of a single point particle of mass ''{{mvar| m''}} with no internal degrees of freedom moving in a [[pseudo-Riemannian]] spacetime manifold '' {{mvar|S''}} with [[metric tensor|metric]] '''g'''. Assume also that the parameter {{mvar|τ}} describing the trajectory of the particle is arbitrary (i.e. we insist upon [[Parametric curve#Reparametrization and equivalence relation|reparametrization invariance]]). Then, its [[symplectic manifold|symplectic space]] is the [[cotangent bundle]] ''T*S'' with the canonical symplectic form {{mvar|ω}}. If we coordinatize ''T'' * ''S'' by its position ''x'' in the base manifold ''S'' and its position within the cotangent space '''p''', then we have a constraint

If we coordinatize ''T'' * ''S'' by its position {{mvar|x}} in the base manifold {{mvar|S}} and its position within the cotangent space '''p''', then we have a constraint

:''f'' = ''m''² −'''g'''(''x'')^−1('''p''','''p''') = 0 .

The Hamiltonian ''{{mvar|H''}} is, surprisingly enough, '' {{mvar|H''}} = 0. In light of the observation that the Hamiltonian is only defined up to the equivalence class of smooth functions agreeing on the constrained subspace, we can use a new Hamiltonian {{mvar|H}} '= {{mvar|f}} instead. Then, we have the interesting case where the Hamiltonian is the same as a constraint! See [[Hamiltonian constraint]] for more details.

Consider now the case of a [[Yang–Mills theory]] for a real [[simple Lie algebra]] ''L'' (with a [[negative definite]] [[Killing form]] η) [[minimally coupled]] to a real scalar field σ, which transforms as an [[orthogonal representation]] ρ with the underlying vector space ''V'' under ''L'' in (''d'' − 1) + 1 [[Minkowski spacetime]]. For l in ''L'', we write

:ρ(l)[σ]

Consider now the case of a [[Yang–Mills theory]] for a real [[simple Lie algebra]] '' {{mvar|L''}} (with a [[negative definite]] [[Killing form]] {{mvar|η}}) [[minimally coupled]] to a real scalar field {{mvar|σ}}, which transforms as an [[orthogonal representation]] {{mvar|ρ}} with the underlying vector space '' {{mvar|V''}} under '' {{mvar|L''}} in ('' {{mvar|d''}} − 1) + 1 [[Minkowski spacetime]]. For {{mvar|l}} in '' {{mvar|L''}}, we write

:{{math|''ρ(l)[σ]''}}

as

:{{math|''l[σ]''}}

for simplicity. Let '''A''' be the '' {{mvar|L''}}-valued [[connection form]] of the theory. Note that the '''A''' here differs from the '''A''' used by physicists by a factor of '' {{mvar|i''}} and '' {{mvar|g''}}. This agrees with the mathematician's convention. The action ''S'' is given by

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

:l[σ]

for simplicity. Let '''A''' be the ''L''-valued [[connection form]] of the theory. Note that the '''A''' here differs from the '''A''' used by physicists by a factor of ''i'' and ''g''. This agrees with the mathematician's convention. The action ''S'' is given by

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

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

:<math>d\bold{A}+\bold{A}\wedge\bold{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;]

and {{mvar|α}} is the orthogonal form for {{mvar|ρ}}.

<!--I hope I have all the signs and factors right. I can't guarantee it.-->

What is the Hamiltonian version of this model? Well, first, we have to split '''A''' noncovariantly into a time component {{mvar|φ}} and a spatial part <math>\vec{A}</math>. Then, the resulting symplectic space has the conjugate variables {{mvar|σ}}, {{math|''π_σ''}} (taking values in the underlying vector space of <math>\bar{\rho}</math>, the dual rep of {{mvar|ρ}}), <math>\vec{A}</math>, <math>\vec{\pi}_A</math>, φ and π_φ. for each spatial point, we have the constraints, π_φ=0 and the [[Gaussian constraint]]

:<math>\vec{D}\cdot\vec{\pi}_A-\rho'(\pi_\sigma,\sigma)=0</math>

where since {{mvar|ρ}} is an [[intertwiner]]

:<math>\rho:L\otimes V\rightarrow V</math>,

{{mvar|ρ}} ' is the dualized intertwiner

:<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(\bold{B}\cdot \bold{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 canmay drop them.

==Second class constraints==