First-class constraint: Difference between revisions

Revision as of 02:07, 26 February 2014

Revision as of 17:49, 9 March 2014

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Suppose we have some constraints

:<math> f_i(x)=0, </math>

for ''n'' smooth functions

:<math>\{ f_i \}_{i= 1}^n</math>

These will only be defined [[chart (topology)|chartwise]] in general. Suppose that everywhere on the constrained set, the ''n'' derivatives of the ''n'' functions are all [[linearly independent]] and also that the [[Poisson bracket]]s

:<math>\{f_i,f_j\}</math>

and

:<math>\{f_i,H\}</math>

all vanish on the constrained subspace. This means we can write

:<math>\{f_i,f_j\}=\sum_k c_{ij}^k f_k</math>

for some smooth functions

:<math>c_{ij}^k</math>

(there is a theorem showing this) and

:<math>\{f_i,H\}=\sum_j v_i^j f_j</math>

for some smooth functions

:<math>v_i^j</math>.

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==Geometric theory==

For a more elegant way, suppose given a [[vector bundle]] over M, with ''n''-dimensional [[fiber]] ''V''. Equip this vector bundle with a [[connection form|connection]]. Suppose too we have a [[smooth section]] ''f'' of this bundle.

Then the [[covariant derivative]] of ''f'' with respect to the connection is a smooth [[linear map]] Δ''f'' from the [[tangent bundle]] ''TM'' to ''V'', which preserves the [[base point]]. Assume this linear map is right [[invertible]] (i.e. there exists a linear map ''g'' such that (Δ''f'')''g'' is the [[identity function|identity map]]) for all the fibers at the zeros of ''f''. Then, according to the [[implicit function theorem]], the subspace of zeros of ''f'' is a [[submanifold]].

The ordinary [[Poisson bracket]] is only defined over <math>C^{\infty}(M)</math>, the space of smooth functions over ''M''. However, using the connection, we can extend it to the space of smooth sections of ''f'' if we work with the [[algebra bundle]] with the [[graded algebra]] of ''V''-tensors as fibers. Assume also that under this Poisson bracket,

:{ ''f'', ''f'' } = 0

(note that it's not true that

:{ ''g'', ''g'' } = 0

in general for this "extended Poisson bracket" anymore) and

:{ ''f'', ''H'' } = 0

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What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other '''on''' the constrained subspace; or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constraint flows all bring the point to another point on the constrained subspace.

Since we wish to restrict ourselves to the constrained subspace only, this suggests that the Hamiltonian, or any other physical [[observable]], should only be defined on that subspace. Equivalently, we can look at the [[equivalence class]] of smooth functions over the symplectic manifold, which agree on the constrained subspace (the [[quotient algebra]] by the [[Ideal (ring theory)|ideal]] generated by the ''f'''s, in other words).

The catch is, the Hamiltonian flows on the constrained subspace depend on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this.

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Look at the [[orbit (group theory)|orbits]] of the constrained subspace under the action of the [[symplectic flow]]s generated by the ''f'''s. This gives a local [[foliation]] of the subspace because it satisfies [[integrability condition]]s ([[Frobenius theorem (differential topology)|Frobenius theorem]]). It turns out if we start with two different points on a same orbit on the constrained subspace and evolve both of them under two different Hamiltonians, respectively, which agree on the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows will always lie in the same orbit at equal times. It also turns out if we have two smooth functions ''A''₁ and ''B''₁, which are constant over orbits at least on the constrained subspace (i.e. physical observables) (i.e. {A₁,f}={B₁,f}=0 over the constrained subspace)and another two A₂ and B₂, which are also constant over orbits such that A₁ and B₁ agrees with A₂ and B₂ respectively over the restrained subspace, then their Poisson brackets {A₁, B₁} and {A₂, B₂} are also constant over orbits and agree over the constrained subspace.

In general, one cannot rule out "[[ergodic]]" flows (which basically means that an orbit is dense in some open set), or "subergodic" flows (which an orbit dense in some submanifold of dimension greater than the orbit's dimension). We can't have [[self-intersecting]] orbits.

For most "practical" applications of first-class constraints, we do not see such complications: the [[quotient space]] of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a [[differentiable manifold]], which can be turned into a [[symplectic manifold]] by projecting the [[symplectic form]] of M onto it (this can be shown to be [[well defined]]). In light of the observation about physical observables mentioned earlier, we can work with this more "physical" smaller symplectic manifold, but with 2n fewer dimensions.

In general, the quotient space is a bit "nasty" to work with when doing concrete calculations (not to mention nonlocal when working with [[diffeomorphism constraint]]s), so what is usually done instead is something similar. Note that the restricted submanifold is a [[bundle]] (but not a [[fiber bundle]] in general) over the quotient manifold. So, instead of working with the quotient manifold, we can work with a [[Section (category theory)|section]] of the bundle instead. This is called [[gauge fixing]].

The ''major'' problem is this bundle might not have a [[global section]] in general. This is where the "problem" of [[global anomaly|global anomalies]] comes in, for example. See [[Gribov ambiguity]]. This is a flaw in quantizing [[gauge theory|gauge theories]] many physicists overlooked.

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==Examples==

Look at the dynamics of a single point particle of mass ''m'' with no internal degrees of freedom moving in a [[pseudo-Riemannian]] spacetime manifold ''S'' with [[metric tensor|metric]] '''g'''. Assume also that the parameter τ describing the trajectory of the particle is arbitrary (i.e. we insist upon [[Parametric_curveParametric curve#Reparametrization_and_equivalence_relationReparametrization and equivalence relation|reparametrization invariance]]). Then, its [[symplectic space]] is the [[cotangent bundle]] T*S with the canonical symplectic form 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

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

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The Hamiltonian ''H'' is, surprisingly enough, ''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 H'=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-MillsYang–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'' &minus; 1) + 1 [[Minkowski spacetime]]. For l in ''L'', we write

:&rho;(l)[&sigma;]

as

:l[&sigma;]

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

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

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

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

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''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 φ and a spatial part <math>\vec{A}</math>. Then, the resulting symplectic space has the conjugate variables σ, π_σ (taking values in the underlying vector space of <math>\bar{\rho}</math>, the dual rep of 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 ρ is an [[intertwiner]]

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

ρ' is the dualized intertwiner

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

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:<math>H=\vec{p}\cdot\dot{\vec{r}}+p_\lambda \dot{\lambda}-L=\frac{p^2}{2m}+p_\lambda \dot{\lambda}+mgz-\frac{\lambda}{2}(r^2-R^2)</math>.

We cannot eliminate <math>\dot{\lambda}</math> at this stage yet. We are here treating <math>\dot{\lambda}</math> as a shorthand for a function of the [[symplectic space]] which we have yet to determine and ''not'' an independent variable. For notational consistency, define <math>u_1=\dot{\lambda}</math> from now on. The above Hamiltonian with the {{math|''p''_''λ''}} term is the "naive Hamiltonian". Note that since, on-shell, the constraint must be satisfied, one cannot distinguish, on-shell, between the naive Hamiltonian and the above Hamiltonian with the undetermined coefficient, <math>\dot{\lambda}=u_1</math>.

We have the [[primary constraint]]

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We require, on the grounds of consistency, that the [[Poisson bracket]] of all the constraints with the Hamiltonian vanish at the constrained subspace. In other words, the constraints must not evolve in time if they are going to be identically zero along the equations of motion.

From this consistency condition, we immediately get the [[First_class_constraintsFirst class constraints#Constrained_Hamiltonian_dynamics_from_a_Lagrangian_gauge_theoryConstrained Hamiltonian dynamics from a Lagrangian gauge theory|secondary constraint]]

:{{math|''r''²−''R''²{{=}}0}} .

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Also see

* {{cite doi|10.1088/0143-0807/4/1/003|noedit}}

* {{cite doi| 10.1103/PhysRevD.42.2049|noedit}}

{{DEFAULTSORT:First Class Constraint}}

[[Category:Classical mechanics]]