The hamiltonian formalism and the hamiltonian H in special relativity has some issues with the definition. In the case of the free particle one possible definition, not completely covariant, is the relativistic energy
There are two others interesting scalars in classical relativistic theories. They are the lagrangian L and the action functional S. The lagrangian is obtained through a Legendre transformation from the hamiltonian:
From the hamiltonian, we get the velocity using the so-called hamiltonian equation:
The action functional is the time integral of the lagrangian:
However, let me point out that the above hamiltonian in SR has some difficulties in gauge field theories. Indeed, it is quite easy to derive that a more careful and reasonable election for the hamiltonian in SR should be zero!
In the case of the free relativistic particle, we obtain
Using the relation between time and proper time (the time dilation formula):
direct substitution provides
And defining the infinitesimal proper length in spacetime as , we get the simple and wonderful result:
Sometimes, the covariant lagrangian for the free particle is also obtained from the following argument. The proper length is defined as
The invariant in spacetime is related with the proper time in this way:
Thus, dividing by
and the free coordinate action for the free particle would be:
Note, that since the election of time “t” is “free”, we can choose to obtain the generally covariant free action:
Remark: the (rest) mass is the “coupling” constant for the free particle proper lenght to guess the free lagrangian
Now, we can see from this covariant action that the relativistic hamiltonian should be a feynmanity! From the equations of motion,
The covariant hamiltonian , different from H, can be build in the following way:
The meaning of this result is hidden in the the next identity ( Noether identity or “hamiltonian constraint” in some contexts):
This strange fact that in SR, a feynmanity as the hamiltonian, is related to the Noether identity for the free relativistic lagrangian, indeed, a consequence of the hamiltonian constraint and the so-called reparametrization invariance . Note, in addition, that the free relativistic particle would also be invariant under diffeomorphisms if we were to make the metric space-time dependent, i.e., if we make the substitution . This last result is useful and important in general relativity, but we will not discuss it further in this moment. In summary, from the two possible hamiltonian in special relativity
the natural and more elegant (due to covariance/invariance) is the second one. Moreover, the free particle lagrangian and action are:
Remark: The true covariant lagrangian dynamics in SR is a “constrained” dynamics, i.e., dynamics where we are undetermined. There are more variables that equations as a result of a large set of symmetries ( reparametrization invariance and, in the case of local metrics, we also find diffeomorphism invarince).
The dynamical equations of motion, for a first order lagrangian (e.g., the free particle we have studied here), read for the lagrangian formalism:
By the other hand, for the hamiltonian formalism, dynamical equations are:
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