First-class constraint: Difference between revisions

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{{distinguish|Primary constraint}}

{{Mainbroader|Dirac bracket}}

AIn [[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 |accessdateaccess-date=29 May 2018|format=PDF |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==

Consider a [[symplecticPoisson manifold]] ''M'' with a [[smooth function|smooth]] Hamiltonian over it (for field theories, ''M'' would be infinite-dimensional).

Suppose we have some constraints

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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]]).

==Geometric theory==

For a more elegant way, suppose given a [[vector bundle]] over <math>\mathcal M</math>, with <math>n</math>-dimensional [[Fiber (mathematics)|fiber]] <math>V</math>. Equip this vector bundle with a [[connection form|connection]]. Suppose too we have a [[Section (fiber bundle)|smooth section]] {{mvar|f}} of this bundle.

Then the [[covariant derivative]] of {{mvar|f}} with respect to the connection is a smooth [[linear map]] <math>\nabla f</math> from the [[tangent bundle]] <math>T\mathcal M</math> to <math>V</math>, which preserves the [[base point]]. Assume this linear map is right [[invertible]] (i.e. there exists a linear map <math>g</math> such that <math>(\Delta f)g</math> is the [[identity function|identity map]]) for all the fibers at the zeros of {{mvar|f}}. Then, according to the [[implicit function theorem]], the subspace of zeros of {{mvar|f}} is a [[submanifold]].

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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 {{mvar|f}} if we work with the [[algebra bundle]] with the [[graded algebra]] of ''V''-tensors as fibers.

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==

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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 "nasty"difficult 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 (mathematics)|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. SeeA global anomaly is different from the [[Gribov ambiguity]]., Thiswhich is when a flawgauge fixing doesn't work to fix a gauge uniquely, in quantizinga global anomaly, there is no consistent definition of the gauge field. A global anomaly is a barrier to defining a quantum [[gauge theory|gauge theories]] manydiscovered physicistsby Witten in overlooked1980.

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==

Consider 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 [[ParametricParametrization curve(geometry)#Reparametrization and equivalence relationInvariance|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 {{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.

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 {{overset|•|''λ''}} at this stage yet. We are here treating {{overset|•|''λ''}} as a shorthand for a function of the [[symplectic manifold|symplectic space]] which we have yet to determine and ''not'' as an independent variable. For notational consistency, define {{math| ''u''₁ {{=}} {{overset|•|''λ''}} }} 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| {{overset|•|''λ''}} {{=}} ''u''₁}}.

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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|''r''²−''R''²{{=}}0}} .

<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>

By the same reasoning, thisThis constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient {{mvar|u}}_2,. At this point,enlarging the Hamiltonian isto

By the same reasoning, this constraint should be added into the Hamiltonian with an undetermined (not necessarily constant) coefficient {{mvar|u}}₂. At this point, the Hamiltonian is

:<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>

LikewiseSimilarly, from this secondary constraint, we getfind the tertiary constraint,

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

<math>\begin{align}

by demanding, for consistency, that <math>\{r^2-R^2,\, H\}_{PB} = 0</math> on-shell.

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\\

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

\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

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where {{mvar|u}}₁, {{mvar|u}}₂, and {{mvar|u}}₃ are still completely undetermined.

Note that, frequently, all constraints that are found from consistency conditions are referred to as "''secondary constraints"'' and secondary, tertiary, quaternary, etc., constraints are not distinguished.

We keep turning the crank, demanding this new constraint have vanishing [[Poisson bracket]]

The tertiary constraint's consistency condition yields

:<math>

0=\{\vec{p}\cdot\vec{r},\, H\}_{PB} = \frac{p^2}{m} - mgz+ \lambda r^2 -2 u_2 r^2 = 0.

</math>

However, now, this is ''not'' a quaternary constraint, but a condition which fixes one of the undetermined coefficients. In particular, it fixes

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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Before analyzing the Hamiltonian, consider the three constraints,

:<math>

\phi_1varphi_1 = p_\lambda, \quad \phi_2varphi_2 = r^2-R^2, \quad \phi_3varphi_3 = \vec{p}\cdot\vec{r}.

</math>

Note the nontrivial [[Poisson bracket]] structure of the constraints. In particular,

:<math>

\{\phi_2varphi_2, \phi_3varphi_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 | pmid = | pmc = |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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==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 = 55–9 | year = 1983 | pmidissue = | pmc =1 |bibcode = 1983EJPh....4....5F | s2cid = 250845310 }}

* {{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 = 20492049–2056 | year = 1990 | pmid = | pmc =10013054 |bibcode = 1990PhRvD..42.2049H }}

{{DEFAULTSORT:First Class Constraint}}

[[Category:Classical mechanics]]

[[Category:Theoretical physics]]