# Odd number of second class constraints (!)

For my thesis, I have calculated the constraints for a system using Dirac method of constraint analysis. The problem is I got odd number of second class constraints (!), which gives me unusual numbers of degrees of freedom in phase space. I might have made some mistakes.

Is there any other method beside Dirac, to analyze the constraints of the system?

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## 3 Answers

If the constrained Hamiltonian system has a finite number of real degrees of freedom$^1$, and if all the constraints are regular, then it is mathematically impossible to have an odd number of second-class constraints. (The proof is very similar to the reason why a symplectic manifold or vector space must be even-dimensional.)

Perhaps OP is actually considering a constrained Hamiltonian field theory with an infinite number of degree of freedom and an infinite number of second-class constraints? (Typically this happens because all the fields, say a position field $\phi(\vec{x},t)$ and a momentum field $\pi(\vec{x},t)$, are labeled by a continuous index, namely the space point $\vec{x}$). In that case, it does not make sense to label $\infty$ as an odd number.

Example$^2$: A typical example of a second-class constraints in 1+1 dimension field theory with canonical equal-time Poisson brackets $$\tag{1} \{\phi(x,t),\pi(y,t)\}~=~ \delta(x-y),$$ $$\tag{2} \{\phi(x,t),\phi(y,t)\}~=~0,$$ $$\tag{3} \{\pi(x,t),\pi(y,t)\}~=~0,$$ is

$$\tag{4} \chi(x,t)~:=~\pi(x,t) -\partial_x\phi(x,t).$$

Naively one may think of (4) as a single (i.e. odd!) second-class constraint, but it is really infinitely many second-class constraints labeled by the position $x$. Their equal-time Poisson brackets are

$$\tag{5} \Delta(x,y)~:=~\{\chi(x,t),\chi(y,t)\}~=~ 2 \delta^{\prime}(x-y)$$

with a formal$^3$ inverse

$$\tag{6} \Delta^{-1}(x,y) ~=~ \frac{1}{4}{\rm sgn}(x-y).$$

For another related example of second-class constraints in Hamiltonian field theory, see also e.g. this Phys.SE answer.

$^1$ The definition of degrees of freedom (d.o.f.) is e.g. discussed in this Phys.SE post. (Note that there is also a field-theoretic notion of d.o.f., which is different. E.g. in pure QED in 3+1 dimensions, the photon has 2 physical polarizations, so one would say that pure QED has 2 physical d.o.f., etc. This is not the notion of d.o.f, that I'm considering here. If OP is counting field-theoretic d.o.f., there is no reason to be surprised to meet an odd number, cf. the Example.)

$^2$ This example is sometimes referred to as a chiral/self-dual boson in 1+1 dimensions.

$^3$ One should impose appropriate boundary conditions at $|x| \to \infty$.

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The fields (I am working with) have finite number of DOF of themselves separately. My work was to analyze how they react if they all interact in the same field. In this case, is it possible to have infinite number of DOF? – aries0152 Oct 2 '12 at 8:59
@aries0152: If you would like a more focused response about your system of second-class constraints, you would have to display the explicit form in the question. – Qmechanic♦ Oct 2 '12 at 9:53
– aries0152 Oct 2 '12 at 12:30
Right, that paper considers field theory, i.e. infinitely many d.o.f. – Qmechanic♦ Oct 2 '12 at 13:37
So with infinite number of dof, does the commutation chain breaks in a point (Like $\dot \phi=0$) ? Or it just continue producing new constraints (second class) ? – aries0152 Oct 2 '12 at 17:32
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