Dirac brackets and gauge in special relativity.

[raul.cuesta]

Hello,
It's well known that the action for a relativistic point particle is:
$$S=-m\int d\tau\left(-\dot{x}^2\right)^{1/2}$$
the canonical momentum is

$$p_{\mu}= \frac{m\dot{x}_{\mu}}{\left(-\dot{x}^2\right)^{1/2}}.$$

This action is invariant under reparametrizations of $$\tau$$, then its canonical Hamiltonian vanishes and we have a primary constraint of first class:
$$\varphi_1=p^{2}+m^{2}\approx 0.$$

Then, in order to eliminate all arbitrary funtions of the system we can use the gauge freedom in the action, this is done by imposing a second constraint $$\varphi_2$$ such that $$\varphi_1$$ becomes second class and now we can use the Dirac brackets to work out the problem.

My question is: Is it valid if I ask the Dirac brackets to be

$$\left\{x^{\mu},p^{\nu}\right\}_{D}=\eta^{\mu\nu},$$
$$\left\{x^{\mu},x^{\nu}\right\}_{D}=\left\{p^{\mu},p^{\nu}\right\}_{D}=0,$$

and then I try to find the conditions on $$\varphi_2$$ and finally work with this brackets?

 

[dextercioby]

Hi, actually the form of the Dirac brackets is always derived, either for a first class or second class constrained system. See more on this matter in Henneaux's book. And btw, the free relativistic particle is easier to study/quantize in the einbein formulation.

 

[raul.cuesta]

Hello again,
I know how to calculate the Dirac brackets, my question is about the gauge fixation. Normaly we choose the form of the constraint $$\varphi_2$$ and then we calculate the Dirac brackets. What I want to know, is if this is valid:
1.-Impose the Dirac brackets to be:
$$\left\{x^{\mu},p^{\nu}\right\}_D=\eta^{\mu\nu}$$
$$\left\{x^{\mu},p^{\nu}\right\}_D=0$$
$$\left\{x^{\mu},p^{\nu}\right\}_D=0$$,
2.-Find restrictions over a posible second-class constraint $$\varphi_2$$ in order to obtain the above brackets. For example:
$$\left\{x^{\mu},p^{\nu}\right\}_D=\eta^{\mu\nu}=\eta^{\mu\nu}-\left\{x^{\mu},\varphi_{a}\right\}C^{-1}_{ab}\left\{\varphi_{b},p^{\nu}\right\},$$
where $$a,b=1,2$$, this is true if:
$$p^{\mu}\left\{p^{2},\varphi_2\right\}\left\{\varphi_2,p^{\nu}\right\}=0,$$
3.-Maybe find who is $$\varphi_2$$ and/or simply work the theory with the brackets in 1.

Greetings!