Is the Hamiltonian Equation of Motion Gauge Dependent?

In summary, the conversation discusses the invariance of the Euler-Lagrange equations under gauge transformations and the construction of the Hamiltonian. The participants also discuss the gauge dependence of the Hamiltonian and the physical quantity of momentum. The resolution is that while the Hamiltonian is gauge-dependent, the equation solutions are gauge-invariant and the physical quantity of momentum is P = p + eA.
  • #1
evilcman
41
2
If we consider the following Lagrangian:
L = m * (dot x)^2 / 2 - e A dot x - e phi
with A the vectorpotential and phithe scalar potential, then
the Euler-Lagrange equations, reduce to the known formula
of Lorentz-force:
m ddot x = e dot x X B + e E
know, this equation is invariant under gauge transformations:
A->A + nabla F
phi -> phi - parcd phi / parcd t
Know this is great, but if we construct the Hamiltonian with the
usual Legendre transform we will get
H = (p+eA)^2 / 2m + e phi
My problem with this is, if a write the (Hamiltonian) equation of
motion now, it seems gauge dependent:
dot x = (p + eA)/m

Isn't this a problem? What's the resolution?
 
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  • #2
A Hamiltonian is always gauge-dependent. Remember the arbitrary constant C that can be added to the scalar potential? It is of the same nature.
 
  • #3
The gauge dependence of the Hamiltonian is not my problem. My problem is that, from these equations of motion, the velocity, which is something that should be mesurable, also seems to be gauge dependent...
 
  • #4
No, the equation solutions are gauge invariant. Moreover, the equations themselves, expressed via EM filed tensions, are gauge invariant.
 
  • #5
I don't see why the solutions should be gauge independent when A(and not its curl or something) is explicitly in the equations of motion...
 
  • #6
If you speak of velocity equation, it contains also p which is gauge-dependent. Write the equation for acceleration d2x/dt2. You will see that this equation will contain the electric and magnetic fields rather than potentials.
 
  • #7
You are right. And if I understand it correctly p is no longer a physical quantity, only p+eA is.
 
  • #8
Yes, there are two momenta: a canonically conjugated p (a formal mathematical construction) and a kinetic P = p + eA (physical quantity).
 
Last edited:

1. What is an Electrodynamic Gauge Transform?

An Electrodynamic Gauge Transform is a mathematical concept used in theoretical physics and electromagnetic theory. It refers to a set of equations that describe the relationship between the electromagnetic potentials and fields of a system. It is used to simplify and analyze complex electromagnetic systems.

2. How does an Electrodynamic Gauge Transform work?

The Electrodynamic Gauge Transform is based on the principle of gauge invariance, which states that the electromagnetic potentials can be changed without affecting the physical results of the system. The transform allows for a change in the gauge of the potentials, while keeping the fields unchanged. This allows for a simpler and more manageable description of the system.

3. What are the applications of the Electrodynamic Gauge Transform?

The Electrodynamic Gauge Transform has many applications in theoretical physics, particularly in quantum field theory and electromagnetism. It is used to study and analyze complex electromagnetic systems, such as electromagnetic waves, quantum electrodynamics, and more. It is also used in the development of advanced technologies, such as particle accelerators and advanced communication systems.

4. Who developed the concept of Electrodynamic Gauge Transform?

The concept of Electrodynamic Gauge Transform was first introduced by the physicist Hermann Weyl in 1918. It was later developed and further refined by other physicists, including Albert Einstein, Wolfgang Pauli, and Julian Schwinger.

5. What is the difference between an Electrodynamic Gauge Transform and a Gauge Transformation?

An Electrodynamic Gauge Transform is a specific type of gauge transformation that is used in electromagnetic theory. A gauge transformation, on the other hand, is a more general concept that can be applied to other physical systems as well. The difference lies in the type of system being described and the specific equations used in the transformation.

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