- #1
raul.cuesta
- 3
- 0
Hello,
It's well known that the action for a relativistic point particle is:
[tex]
S=-m\int d\tau\left(-\dot{x}^2\right)^{1/2}
[/tex]
the canonical momentum is
[tex]
p_{\mu}= \frac{m\dot{x}_{\mu}}{\left(-\dot{x}^2\right)^{1/2}}.
[/tex]
This action is invariant under reparametrizations of [tex]\tau[/tex], then its canonical Hamiltonian vanishes and we have a primary constraint of first class:
[tex]
\varphi_1=p^{2}+m^{2}\approx 0.
[/tex]
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 [tex]\varphi_2[/tex] such that [tex]\varphi_1[/tex] 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
[tex]
\left\{x^{\mu},p^{\nu}\right\}_{D}=\eta^{\mu\nu},
[/tex]
[tex]
\left\{x^{\mu},x^{\nu}\right\}_{D}=\left\{p^{\mu},p^{\nu}\right\}_{D}=0,
[/tex]
and then I try to find the conditions on [tex]\varphi_2[/tex] and finally work with this brackets?
It's well known that the action for a relativistic point particle is:
[tex]
S=-m\int d\tau\left(-\dot{x}^2\right)^{1/2}
[/tex]
the canonical momentum is
[tex]
p_{\mu}= \frac{m\dot{x}_{\mu}}{\left(-\dot{x}^2\right)^{1/2}}.
[/tex]
This action is invariant under reparametrizations of [tex]\tau[/tex], then its canonical Hamiltonian vanishes and we have a primary constraint of first class:
[tex]
\varphi_1=p^{2}+m^{2}\approx 0.
[/tex]
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 [tex]\varphi_2[/tex] such that [tex]\varphi_1[/tex] 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
[tex]
\left\{x^{\mu},p^{\nu}\right\}_{D}=\eta^{\mu\nu},
[/tex]
[tex]
\left\{x^{\mu},x^{\nu}\right\}_{D}=\left\{p^{\mu},p^{\nu}\right\}_{D}=0,
[/tex]
and then I try to find the conditions on [tex]\varphi_2[/tex] and finally work with this brackets?