- #1
Geometry_dude
- 112
- 20
At the moment I am trying to understand classical field theory and there's a conceptual problem I encountered, which bothers me a lot and I don't seem to be able to resolve the issue. When making the step to classical field theory, many texts start as follows:
First they recall the/a action in mass point mechanics and write something like this
$$S = \int L \, d t \, ,$$
then they say that we don't want to look at particles anymore but at (say scalar) fields ##\phi## and introduce the Lagrangian density ##\mathcal L## as a function of the "field variables" ##\phi## and its derivatives, say ##\partial_t \phi, \partial_x \phi## as follows
$$L = \int_{- \infty}^{\infty} \mathcal L \, d x \, . $$
Now here's my problem: This equation is rubbish, plain mathematical nonsense.
As Ben Niehoff already clarified in a similar question I asked some time ago ( https://www.physicsforums.com/showthread.php?t=751858 ),
there's an action for curves ##\gamma## and an action for fields ##\phi## and the two are, at least on a mathematical level, two separate concepts. If we want to be a bit more rigorous, we thus write the mass point action like
$$S(\gamma, \tau_1, \tau_2) = \int_{\tau_1}^{\tau_2} L ( \gamma, \dot \gamma) \, d \tau$$
and the field action of a (real) scalar field ##\phi## in analogy
$$\mathcal S (\phi, \phi_\text{boundary}) = \int_{\text{interior spacetime}} \mathcal L(\phi, d \phi)$$
where ##\mathcal L(\phi, d \phi)## is now an actual density on the spacetime ( http://en.wikipedia.org/wiki/Density_on_a_manifold ).
This clarifies why the equation I was talking about is nonsense: ##L## and ##\mathcal L## have two distinct inputs, i.e. live on different spaces and the equation is not invariant under coordinate change.
So my questions are:
1) What is the correct step/transition from Mass Point Mechanics to Field Theory?
2) Why do we only consider first order derivatives of the field in the Lagrangian density?
3) How do I get to the Hamiltonian formalism in the field theory in a coordinate independent manner?
First they recall the/a action in mass point mechanics and write something like this
$$S = \int L \, d t \, ,$$
then they say that we don't want to look at particles anymore but at (say scalar) fields ##\phi## and introduce the Lagrangian density ##\mathcal L## as a function of the "field variables" ##\phi## and its derivatives, say ##\partial_t \phi, \partial_x \phi## as follows
$$L = \int_{- \infty}^{\infty} \mathcal L \, d x \, . $$
Now here's my problem: This equation is rubbish, plain mathematical nonsense.
As Ben Niehoff already clarified in a similar question I asked some time ago ( https://www.physicsforums.com/showthread.php?t=751858 ),
there's an action for curves ##\gamma## and an action for fields ##\phi## and the two are, at least on a mathematical level, two separate concepts. If we want to be a bit more rigorous, we thus write the mass point action like
$$S(\gamma, \tau_1, \tau_2) = \int_{\tau_1}^{\tau_2} L ( \gamma, \dot \gamma) \, d \tau$$
and the field action of a (real) scalar field ##\phi## in analogy
$$\mathcal S (\phi, \phi_\text{boundary}) = \int_{\text{interior spacetime}} \mathcal L(\phi, d \phi)$$
where ##\mathcal L(\phi, d \phi)## is now an actual density on the spacetime ( http://en.wikipedia.org/wiki/Density_on_a_manifold ).
This clarifies why the equation I was talking about is nonsense: ##L## and ##\mathcal L## have two distinct inputs, i.e. live on different spaces and the equation is not invariant under coordinate change.
So my questions are:
1) What is the correct step/transition from Mass Point Mechanics to Field Theory?
2) Why do we only consider first order derivatives of the field in the Lagrangian density?
3) How do I get to the Hamiltonian formalism in the field theory in a coordinate independent manner?