First-class constraint: Difference between revisions

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{{distinguish|Primary constraint}}

{{broader|Dirac bracket}}

In [[physics]], a '''first -class constraint''' is a dynamical quantity in a constrained [[Hamiltonian mechanics|Hamiltonian]] system whose [[Poisson bracket]] with all the other constraints vanishes on the '''constraint surface''' in [[phase space]] (the surface implicitly defined by the simultaneous vanishing of all the constraints). To calculate the first -class constraint, one assumes that there are no '''second -class constraints''', or that they have been calculated previously, and their [[Dirac bracket]]s generated.<ref name=FysikSuSePDF>{{cite web |author1=Ingemar Bengtsson, Stockholm University|title=Constrained Hamiltonian Systems |url=http://3dhouse.se/ingemar/Nr13.pdf |publisher=Stockholm University |access-date=29 May 2018 |quote=We start from a Lagrangian <math>L ( q, ̇\dot q ),</math> derive the canonical momenta, postulate the naive Poisso nPoisson brackets, and compute the Hamiltonian. For simplicity, one assumes that no second class constraints occur, or if they do, that they have been dealt with already and the naive brackets replaced with Dirac brackets. There remain a set of constraints [...]}}</ref>

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 vector space|symplectic form]] is degenerate.<ref>{{Citation|title=Generalized Hamiltonian dynamics|year=1950|last1=Dirac|first1=Paul A. M.|author1-link=Paul Dirac|journal=[[Canadian Journal of Mathematics]]|volume=2|pages=129–148|doi=10.4153/CJM-1950-012-1|issn=0008-414X|mr=0043724|s2cid=119748805 |doi-access=free}}</ref><ref>{{Citation|title=Lectures on Quantum Mechanics|url=https://books.google.com/books?id=GVwzb1rZW9kC|year=1964|last1=Dirac|first1=Paul A. M.|series=Belfer Graduate School of Science Monographs Series|volume=2|publisher=Belfer Graduate School of Science, New York| isbn=9780486417134 |mr=2220894}}. Unabridged reprint of original, Dover Publications, New York, NY, 2001. </ref>

The terminology of first- and second -class constraints is confusingly similar to that of [[primary constraint|primary and secondary constraints]], reflecting the manner in which these are generated. 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.

==Poisson brackets==

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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> — −−therethere 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>.

This can be done globally, using a [[partition of unity]]. Then, we say we have an irreducible '''first-class constraint''' (''irreducible'' here is in a different sense from that used in [[representation theory]]).

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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 &minus;'''g'''(''x'')^&minus;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]] {{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}} &minus; 1) + 1 [[Minkowski spacetime]]. For {{mvar|l}} in {{mvar|L}}, we write

:{{math|''&rho;(l)[&sigma;]''}}

as

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{{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(\mathbf{B}\cdot \mathbf{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.

==Second -class constraints==

In a constrained Hamiltonian system, a dynamical quantity is '''second -class''' if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a '''second -class constraint'''.

See [[Dirac bracket]]s for diverse illustrations.

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

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The [[conjugate momentum|conjugate momenta]] are given by

:<math>p_x=m\dot{x}</math>, <math>p_y=m\dot{y}</math>, <math>p_z=m\dot{z}</math>, <math>p_\lambda=0</math> .

Note that we can't determine {{math|{{overset|•|''λ''}}}} from the momenta.

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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 constraints#Constrained Hamiltonian dynamics from a Lagrangian gauge theory|secondary constraint]]

<math>\begin{align}

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\{\varphi_2, \varphi_3\} = 2 r^2 \neq 0.

</math>

The above Poisson bracket does not just fail to vanish off-shell, which might be anticipated, but ''even on-shell it is nonzero''. Therefore, {{math| ''φ''₂}} and {{math| ''φ''₃}} are '''second -class constraints''', while {{math| ''φ''₁}} is a first -class constraint. Note that these constraints satisfy the regularity condition.

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

Effectively, these brackets (illustrated for this spherical surface in the [[Dirac bracket]] article) project the system back onto the constraints surface.

If one then wished to canonically quantize this system, then one need promote the canonical Dirac brackets,<ref>{{Cite journal | last1 = Corrigan | first1 = E. | last2 = Zachos | first2 = C. K. | doi = 10.1016/0370-2693(79)90465-9 | title = Non-local charges for the supersymmetric σ-model | journal = Physics Letters B | volume = 88 | issue = 3–4 | pages = 273 | year = 1979 |bibcode = 1979PhLB...88..273C }}</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 {{mvar|λ}} dropped out of the extended Hamiltonian. Since {{math| ''φ''₁}} 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 {{mvar|λ}}, which has ceased to have any effect on the particle's dynamics. Therefore, that {{mvar|λ}} dropped out of the Hamiltonian, that {{mvar|u}}₁ is undetermined, and that {{math| ''φ''₁}} = ''p_λ'' 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}} as a primary constraint and proceed through the formalism: The result would the elimination of the extraneous {{mvar|λ}} dynamical quantity. However, the example is more edifying in its current form.

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and

:<math>B_{ij} \equiv \frac{\partial A_j}{\partial x_i} - \frac{\partial A_i}{\partial x_j}</math>.

<math>(\vec{A},-\vec{E})</math> and <math>(\phi,\pi)</math> are [[canonical variables]]. The second -class constraints are

:<math>\pi \approx 0</math>

and

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* {{Cite journal | last1 = Homma | first1 = T. | last2 = Inamoto | first2 = T. | last3 = Miyazaki | first3 = T. | doi = 10.1103/PhysRevD.42.2049 | title = Schrödinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space | journal = Physical Review D | volume = 42 | issue = 6 | pages = 2049–2056 | year = 1990 | pmid = 10013054 |bibcode = 1990PhRvD..42.2049H }}

{{DEFAULTSORT:First Class Constraint}}

[[Category:Classical mechanics]]

[[Category:Theoretical physics]]