But the classical electromagnetic tensor is directly included in the Lagrangian, so I don't see how it's not derived from it. |
This doesn't really make much sense as a response to my statement.
I said that "Quantum Mechanics is not derived from Classical Field Theory". That is, the principles of quantum mechanics were/are not developed out of the physics of classical field theory.
Saying that the electromagnetic field tensor* is in the Lagrangian has nothing to do with this.
As an analogy, imagine if I had made the statement that "Darwin did not deduce evolution from medical descriptions of the liver". Your response would be equivalent to "But this evolutionary textbook has a chapter about the liver".
If you start with basic quantum mechanics, the entire theory is based upon the position and momentum of an electron in a classical potential. |
Quantum mechanics doesn't rest on the position and momentum of the electron. It can describe any particle we've currently observed, none are fundamental to the formalism.
QED reduces the degrees of freedom via minimal coupling |
Wow, QED increases the degrees of freedom from conventional quantum mechanics by an infinite amount!
In the quantum mechanical theory of an electron in a potential there are only six degrees of freedom, the spatial coordinates of the spin up and spin down component of the electron (i.e. x,y,z position of each spin).
In QED there is an infinite number of degrees of freedom, a degree of freedom for the value of the electron and photon field at each spacetime point.
Einstein’s field equations add a bit more than the space-time metric |
True, but that has nothing to do with my statement, I never mentioned the metric.
but if you look at the mechanics of QED, the only additions with respect to classical Lagrangian dynamics are spin and probability |
The addition of probability turns it into a completely different theory, described using a completely different branch of mathematics, with entirely different predictions.
Also the classical version of the Lagrangian already has spin.
Probability is the product of the ensemble interpretation of QM and requires nothing beyond classical position and momentum. |
I don't really understand what this means. How does probability "require" classical position and momentum.
The Dirac field is only real in the sense of momentum and position. |
The Dirac field doesn't describe position or momentum, so I don't understand what this means. The Dirac field is an object which fills spacetime and which commonly has excitations known as electrons. Can you explain what your statement means.
Using space-time algebra, the spinor component of the Dirac equation can be directly interpreted as a point spinning around the classical position of an electron. |
A classical spinor can be viewed this way. That is, a single spinor can be viewed this way. However, a spinor field, where you now have a spinor at each point in spacetime can not.
More importantly: You seem to be conflating the Dirac field and the electron in your writings. Electrons are excitations of the Dirac field. For example:
These “hidden” variables determine the position and momentum of a particle, which is depicted by the Dirac field. |
The Dirac Field does not depict or describe an individual electron, they arise as excitations of it.
Quantization is the procedure of constraining something from a continuous set of values to a relatively small discrete set |
That is not the definition used in physics.
So if a particle is a localized field that exists over all points in space, quantization is reducing this continuous set of positions to the classical location |
That sounds more like wave-function collapse.
QFT I believe refers to particles as field condensates, but it is in general the same concept. |
Particles are excitations of the field in QFT.
*For those reading the electromagnetic field tensor is the mathematical object used to describe the electric and magnetic fields.