First-class constraint: Difference between revisions

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In a constrained Hamiltonian system, a dynamical quantity is called a '''first class constraint''' if its Poisson bracket with all the other constraints vanishes on the '''constraint surface''' (the surface implicitly defined by the simultaneous vanishing of all the constraints). A '''second class constraint''' is one that is not first class.

First and second class constraints were introduced by {{harvs|txt|last=Dirac|authorlink=Paul Dirac|year1=1950|loc=p.136|year2=1964|loc2=p.17}} as a way of quantizing mechanical systems such as gauge theories where the symplectic form is degenerate.<ref>{{Citation | last1=Dirac | first1=P. A. M. | author1-link=Paul Dirac | title=Generalized Hamiltonian dynamics | doi=10.4153/CJM-1950-012-1 | id={{MR|0043724}} | year=1950 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=2 | pages=129–148}}</ref>

*<ref>{{Citation | last1=Dirac | first1=Paul A. M. | title=Lectures on quantum mechanics | url=http://books.google.com/books?id=GVwzb1rZW9kC | publisher=Belfer Graduate School of Science, New York | series=Belfer Graduate School of Science Monographs Series | id={{MR|2220894}} Reprinted by Dover in 2001. | year=1964 | volume=2}}</ref>

The terminology of first and second class constraints is confusingly similar to that of [[primary constraint|primary and secondary constraints]]. These divisions are independent: both first and second class constraints can be either primary or secondary, so this gives altogether four different classes of constraints.

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===An example: a particle confined to a sphere===

{{Disputeabout|"The second class constraints and Hamiltonian given in this example"|date=March 2009}}

Before going on to the general theory, let's look at a specific example step by step to motivate the general analysis.

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</math>

And from the secondary constraint, we get the tertiary constraint,

<math>\vec{p}\cdot\vec{r}=0</math>,

by demanding, on the grounds offor consistency, that <math>\{r^2-R^2,\, H\}_{PB} = 0</math> on-shell. Again, one should add this constraint into the Hamiltonian, since on-shell no one can tell the difference. Therefore, so far, the Hamiltonian looks like

by demanding on the grounds of consistency that <math>\{r^2-R^2,\, H\}_{PB} = 0</math> on-shell. Again, one should add this constraint into the Hamiltonian since on-shell no one can tell the difference. Therefore, so far, the Hamiltonian looks like

:<math>

H = \frac{p^2}{2m} + mgz - \frac{\lambda}{2}(r^2-R^2) + u_1 p_\lambda + u_2 (r^2-R^2) + u_3 \vec{p}\cdot\vec{r},

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</math>

At this point there are ''no more constraints or consistency conditions'' to check.

Putting it all together,

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Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace. But [[Paul Dirac|Dirac]] noticed that we can turn the underlying [[differential manifold]] of the [[symplectic space]] into a [[Poisson manifold]] using a different bracket, called the [[Dirac bracket]], such that the Dirac bracket of any (smooth) function with any of the second class constraints always vanishes and a couple of other nice properties.

If one wanted to canonically quantize this system, then, one needs to promote the canonical Dirac brackets<ref>{{cite doi|10.1016/0370-2693(79)90465-9|noedit}}</ref> not the canonical Poisson brackets to commutation relations.

Examination of the above Hamiltonian shows a number of interesting things happening. One thing to note is that on-shell when the constraints are satisfied the extended Hamiltonian is identical to the naive Hamiltonian, as required. Also, note that <math>\lambda</math> dropped out of the extended Hamiltonian. Since <math>\phi_1</math> is a first class primary constraint it should be interpreted as a generator of a gauge transformation. The gauge freedom is the freedom to choose <math>\lambda</math> which has ceased to have any effect on the particle's dynamics. Therefore, that <math>\lambda</math> dropped out of the Hamiltonian, that <math>u_1</math> is undetermined, and that <math>\phi_1 = p_\lambda</math> is first class, are all closely interrelated.

Note that it would be more natural not to start with a Lagrangian with a Lagrange multiplier, but instead take <math>r^2-R^2</math> as a primary constraint and proceed through the formalism. The result would the elimination of the extraneous <math>\lambda</math>{{mvar|λ}} dynamical quantity. Perhaps, the example is more edifying in its current form.

===Example: Proca action===

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The Hamiltonian is given by

:<math>H = \int d^dx \left[ \frac{1}{2}E^2 + \frac{1}{4}B_{ij}B_{ij} - \pi\nabla\cdot\vec{A} + \vec{E}\cdot\nabla\phi + \frac{m^2}{2}A^2 - \frac{m^2}{2}\phi^2\right]</math>.

1, N. K. Falck and A. C. Hirshfeld, 1983, "Dirac-bracket quantisation of a constrained nonlinear system: the rigid rotator", Eur. J. Phys. 4 p.&nbsp;5. {{doi|10.1088/0143-0807/4/1/003}} (Note that the form of the quantum momentum in this paper is dubious.)

2, T. Homma, T. Inamoto, and T. Miyazaki, 1990, "Schrödinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space ", Phys. Rev. D 42, p.&nbsp;2049. http://prd.aps.org/abstract/PRD/v42/i6/p2049_1. (Tote that the Hamiltonian suggested by the authors from second form of the constraint, (i.e., the time derivative of the <math> f(x)=0 </math> ), is not completely compatible with the formalism of Hamiltonian mechanics.)

==See also==

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==References==

{{reflist}}

1,* N. K. Falck and A. C. Hirshfeld, 1983, "Dirac-bracket quantisation of a constrained nonlinear system: the rigid rotator", Eur. J. Phys. 4 p.&nbsp;5. {{doi|10.1088/0143-0807/4/1/003}} (Note that the form of the quantum momentum in this paper is dubious.)

2,* T. Homma, T. Inamoto, and T. Miyazaki, 1990, "Schrödinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space ", Phys. Rev. D 42, p.&nbsp;2049. http://prd.aps.org/abstract/PRD/v42/i6/p2049_1. (Tote that the Hamiltonian suggested by the authors from second form of the constraint, (i.e., the time derivative of the <math> f(x)=0 </math> ), is not completely compatible with the formalism of Hamiltonian mechanics.)

*{{Citation | last1=Dirac | first1=P. A. M. | author1-link=Paul Dirac | title=Generalized Hamiltonian dynamics | doi=10.4153/CJM-1950-012-1 | id={{MR|0043724}} | year=1950 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=2 | pages=129–148}}

*{{Citation | last1=Dirac | first1=Paul A. M. | title=Lectures on quantum mechanics | url=http://books.google.com/books?id=GVwzb1rZW9kC | publisher=Belfer Graduate School of Science, New York | series=Belfer Graduate School of Science Monographs Series | id={{MR|2220894}} Reprinted by Dover in 2001. | year=1964 | volume=2}}

{{DEFAULTSORT:First Class Constraint}}