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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 |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 Poisson 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==
Consider a [[
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.
:<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>.
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]]).
==Geometric theory==
For a more elegant way, suppose given a [[vector bundle]] over <math>\mathcal M</math>, with
Then the [[covariant derivative]] of
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
Assume also that under this Poisson bracket, <math>\{f,f\}=0</math> (note that it's not true that <math>\{g,g\}=0</math> in general for this "extended Poisson bracket" anymore) and <math>\{f,H\}=0</math> on the submanifold of zeros of {{mvar|f}} (If these brackets also happen to be zero everywhere, then we say the constraints close [[off shell]]). It turns out the right invertibility condition and the commutativity of flows conditions are ''independent'' of the choice of connection. So, we can drop the connection provided we are working solely with the restricted subspace.
==Intuitive meaning==
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 associative algebra|quotient algebra]] by the [[Ideal (ring theory)|ideal]] generated by the
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.
Look at the [[orbit (group theory)|orbits]] of the constrained subspace under the action of the [[Symplectomorphism|symplectic
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.
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For most "practical" applications of first-class constraints, we do not see such complications: the [[Quotient space (topology)|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
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.
What have been described are irreducible first-class constraints. Another complication is that Δf might not be [[right invertible]] on subspaces of the restricted submanifold of [[codimension]] 1 or greater (which violates the stronger assumption stated earlier in this article). This happens, for example in the [[cotetrad]] formulation of [[general relativity]], at the subspace of configurations where the [[cotetrad field]] and the [[connection form]] happen to be zero over some open subset of space. Here, the constraints are the diffeomorphism constraints.
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==Examples==
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''<sup>2</sup> −'''g'''(''x'')<sup>−1</sup>('''p''','''p''') = 0.
The Hamiltonian
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 {{mvar|S}} is given by
:<math>S[\mathbf{A},\sigma]=\int d^dx \frac{1}{4g^2}\eta((\mathbf{g}^{-1}\otimes \mathbf{g}^{-1})(\mathbf{F},\mathbf{F}))+\frac{1}{2}\alpha(\mathbf{g}^{-1}(D\sigma,D\sigma))</math>
where '''g''' is the Minkowski metric, '''F''' is the [[curvature form]]
:<math>d\
(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
:Dσ = dσ − '''A'''[σ]
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 {{vec|''A''}}. Then, the resulting symplectic space has the conjugate variables {{mvar|σ}}, {{math|''π<sub>σ</sub>''}} (taking values in the underlying vector space of <math>\bar{\rho}</math>, the dual rep of {{mvar|ρ}}), {{vec|''A''}}, {{vec|''π''}}<sub>''A''</sub>, ''φ'' and ''π<sub>φ</sub>''. For each spatial point, we have the constraints, ''π<sub>φ</sub>''=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(\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
In a constrained Hamiltonian system, a dynamical quantity is '''second
See
===An example: a particle confined to a sphere===
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Before going on to the general theory, consider a specific example step by step to motivate the general analysis.
Start with the [[action (physics)|action]] describing a [[Newtonian dynamics|Newtonian]] particle of [[mass]] {{mvar|m}} constrained to a spherical surface of radius {{mvar|R}} within a uniform [[gravitational field]] {{mvar|g}}. When one works in Lagrangian mechanics, there are several ways to implement a constraint: one can switch to generalized coordinates that manifestly solve the constraint, or one can use a Lagrange multiplier while retaining the redundant coordinates so constrained.
In this case, the particle is constrained to a sphere, therefore the natural solution would be to use angular coordinates to describe the position of the particle instead of Cartesian and solve (automatically eliminate) the constraint in that way (the first choice). For pedagogical reasons, instead, consider the problem in (redundant) Cartesian coordinates, with a Lagrange multiplier term enforcing the constraint.
The action is given by
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where the last term is the [[Lagrange multiplier]] term enforcing the constraint.
Of course, as indicated, we could have just used different, non-redundant, spherical [[coordinates]] and written it as
:<math>S=\int dt \left[\frac{mR^2}{2}(\dot{\theta}^2+\sin^2(\theta)\dot{\phi}^2)+mgR\cos(\theta)\right]</math>
instead, without extra constraints
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
The [[Hamiltonian mechanics|Hamiltonian]] is given by
:<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
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 [[
<math>\begin{align}
0&=\{H,p_\lambda\}_\text{PB}\\
&=\sum_{i}\frac{\partial H}{\partial q_i}\frac{\partial p_\lambda}{\partial p_i}-\frac{\partial H}{\partial p_i}\frac{\partial p_\lambda}{\partial q_i}\\
&=\frac{\partial H}{\partial \lambda}\\
&=\frac{1}{2}(r^2-R^2)\\
&\Downarrow\\
0&=r^2-R^2
\end{align}</math>
This constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient {{mvar|u}}<sub>2,</sub> enlarging the Hamiltonian to
:<math>
H = \frac{p^2}{2m} + mgz - \frac{\lambda}{2}(r^2-R^2) + u_1 p_\lambda + u_2 (r^2-R^2) ~.
</math>
<math>\begin{align}
0&=\{H,r^2-R^2\}_{PB}\\
&=\{H,x^2\}_{PB}+\{H,y^2\}_{PB}+\{H,z^2\}_{PB}\\
&=\frac{\partial H}{\partial p_x}2x+\frac{\partial H}{\partial p_y}2y+\frac{\partial H}{\partial p_z}2z\\
&=\frac{2}{m}(p_xx+p_yy+p_zz)\\
&\Downarrow\\
0&=\vec p\cdot\vec r
\end{align}</math>
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}~,
</math>
where {{mvar|u}}<
Note that, frequently, all constraints that are found from consistency conditions are referred to as We keep turning the crank, demanding this new constraint have vanishing [[Poisson bracket]]
:<math>
0=\{\vec{p}\cdot\vec{r},\, H\}_{PB} = \frac{p^2}{m} - mgz+ \lambda r^2 -2 u_2 r^2
</math>
We might despair and think that there is no end to this, but because one of the new Lagrange multipliers has shown up, this is not a new constraint, but a condition that fixes the Lagrange multiplier:
:<math>
u_2 = \frac{\lambda}{2} + \frac{1}{r^2}\left(\frac{p^2}{2m}-\frac{1}{2}mgz \right).
</math>
Plugging this into our Hamiltonian gives us (after a little algebra)
<math>
H = \frac{p^2}{2m}(2-\frac{R^2}{r^2}) + \frac{1}{2}mgz(1+\frac{R^2}{r^2})+u_1p_\lambda+u_3\vec p \cdot\vec r
</math>
Now that there are new terms in the Hamiltonian, one should go back and check the consistency conditions for the primary and secondary constraints. The secondary constraint's consistency condition gives
:<math>
\frac{2}{m}\vec{r}\cdot\vec{p} + 2 u_3 r^2 = 0.
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Again, this is ''not'' a new constraint; it only determines that
:<math>
u_3 = -\frac{\vec{r}\cdot\vec{p}}{m r^2}~.
</math>
At this point there are ''no more constraints or consistency conditions to check''!
Putting it all together,
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</math>
Before analyzing the Hamiltonian, consider the three constraints
:<math>
\
</math>
:<math>
\{\
</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''<sub>2</sub>}} and
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''<sub>1</sub>}} 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}}<sub>1</sub> is undetermined, and that {{math| ''φ''<sub>1</sub>}} = ''p<sub>λ</sub>'' 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.
{{see also|Dirac bracket}}
===Example: Proca action===
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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
:<math>\pi \approx 0</math>
and
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==Further reading==
* {{Cite journal | last1 = Falck | first1 = N. K. | last2 = Hirshfeld | first2 = A. C. | doi = 10.1088/0143-0807/4/1/003 | title = Dirac-bracket quantisation of a constrained nonlinear system: The rigid rotator | journal = European Journal of Physics | volume = 4 | pages =
* {{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 =
[[Category:Classical mechanics]]
[[Category:Theoretical physics]]
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