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3

It's because magnetic and electric fields transform into one another depending on your intertial frame, and therefore field lines aren't an invariant intrinsic property of space. In one frame, you could have a region where there's only a uniform magnetic field $\vec B$, so that any stationary charge remains classically at rest. Yet viewed from a frame ...


1

This is because it is assumed that the test charge does not produce any electric field of its own and its magnitude is negligibly small, so it doesn't apply any force on the test charge.


0

The geodesic hypothesis pertains to "test particles", not to fields. However, given that it does pertain to photons as such, and photons are the quanta of the EM field, you could say that, yes, the geodesic hypothesis assumes how the quanta of the EM field evolves in spacetime.


3

OP considers an equations of motion of the form $$\tag{1}\dot{\bf x}~=~{\bf B}({\bf x}),$$ where the vector field ${\bf B}$ is of the form$^1$ $$\tag{2} {\bf B}~=~{\bf \nabla}\times {\bf A}.$$ In other words, ${\bf B}$ is divergence-free $$\tag{3} {\bf \nabla}\cdot {\bf B}~=~0.$$ Eq.(3) is locally eqivalent to eq. (2), cf. Poincare's Lemma. Let ...


0

That a free particle follows a geodesic follows from the principle of least action and taking the action as $$S = \int d\tau \frac{m}{2} g_{\mu\nu} \dot{x}^\nu\dot{x}^\mu$$ (really just the generalization of the action of a free particle as just the kinetic energy). Similarly you can derive the equations of motion for the electromagnetic field from the ...


0

Tong's lecture notes cover this in detail. In particular, your lagrangian is easy to obtain from a complex scalar field that obeys the Klein-Gordon equation in the non-relativistic limit. This lagrangian does lead to a conserved Nöther current of the same form as that of Schrödinger's, but this does not have the interpretation of conserved probability, ...


3

Let us here assume that the classical theory is given by a Hamiltonian (as opposed to a Lagrangian) formulation, so that we have a Poisson bracket $\{\cdot,\cdot\}_{PB}$ (and so that we can discuss whether or not the generators form a Poisson algebra or not). A generic theory will have constraints (and corresponding Lagrange multipliers). The constraints ...


1

In gauge theory, the connection (the potential in the electromagnetic case) is defined over a fiber bundle with fibers in some suitable Lie Group. Yang Mills theories are theories with Lie Group $SU(N)$, while in electromagnetism the group is $U(1)$. So for all these theories, the fiber in the fiber bundle is a Lie Group. The case where I know there is ...


1

but when they are pushed in the same direction, they create what we call an uniform electric field Movement of electrons in the same direction does not create a uniform electric field. Two infinite parallel plates with an electric potential difference between them, and no movement of electrons or other charged particles, would set up a uniform ...


2

QED is an innately relativistic theory since it has massless particles (photons). Thus to take a non-relativistic limit you must remove the photon degree of freedom. This can be done by treating the electromagnetic field as a classical field, i.e. one that isn't quantized. Practically this amounts to only having real (on-shell) photons. The scalar QED ...


1

How can we visualise the magnetic field? Easy draw little lines coming out of one end of a magnet, and follow some rules (like lines must be closed, these lines tend to "repel" each other if possible and do not cross). From this we can tell a lot about the magnet and the field around it. But in reality we usually solve for the magnetic field first and ...


0

First, one would have to define what the north and south pole should be. In practice this is not so clear. Actually, the current loop is the more fundamental concept for producing a magnetic field. E.g., a bar magnet can simplified be imagined as a composition of a huge number of current loops. Their magnetic fields add up to the overall field of the bar ...


2

Consider a theory of fields $\phi:M\to T$ where $M$ is a manifold, and $T$ is a set. In physics, $T$ is often either a vector space or a manifold. We call $M$ the domain of the theory, and we call $T$ the target space. of the theory. We call a function from $M$ to $T$ a field configuration, and the set of all field configurations is denoted $\mathcal F$. ...


1

To find the total force, you can simply add up all the contributions of all bodies. Depending on the distribution or initial/boundary conditions of the bodies, there may be clever tricks to make it easier. If you want to calculate (for example) the orbit of an object in a three+ body problem, you'll have to resort to either approximations or compute it ...


1

In QFT, the Dirac spinor will also be promoted to a field, whose oscillation mode coefficients are creation and annihilation operators. BUT: For the Dirac spinor it is possible to well-define a probablility density and current: $$\rho^\mu \propto \bar\psi (\partial^\mu \psi) - (\partial^\mu \bar \psi) \psi $$ This current's zero component is positive ...



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