# Tag Info

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OP is right. In general, to make the action principle work, we only need to impose boundary conditions (BCs) at the boundary of spacetime. Hence we shouldn't impose eq. (6) in the interior of spacetime. In other words, OP should preferably redo their above analysis using the action rather than the Lagrangian.

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Let's consider a simpler problem: the vibrations of a string. We will describe it with the following Lagrangian density $$\mathscr{L} = \frac{1}{2}\partial_a\phi\partial^a\phi\,.$$ The equation of motion is $$0 = \partial_t^2\phi - \partial_x^2\phi\,.$$ The solution of this equation is any function of the form $$\phi = f(t + x) + g(t - x)\,.$$ We must now ...

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First, for fermion component notation in the Martin's textbook. Forget your notations for a while, and start from the beginning. For Weyl spinors, let me replace the dagger (h.c.) with the bar to avoid clutter (which is quite a common practice). This bar (or dagger) always accompanies the dotted indices, upper or lower, while undotted indices are always ...

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There are two facts to be distinguished. Electrons are what we loosely call particles, so they only ever occur in discrete numbers. Millikan demonstrated the discreteness of charge. Secondly, localised states in general have discrete energies. Examples of these are atomic and molecular states. Free electrons have continuous electrons. So called free ...

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I'll address what I understand to be your question, namely Where is the discreteness of the number of excitations of QFT coming from, even in free propagation that apparently does not involve a potential and the corresponding compactness associated with discreteness? (This is the magic of Fock space often referred to as "second quantization", an ...

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In normal quantum mechanics, we consider an individual particle, turn its momentum and position into vector operators $\hat{P}_i$ and $\hat{X}_i$, and enforce the canonical commutation relations $[\hat{X}_i,\hat{P}_j]=i\hbar\delta_{ij}$. In Quantum Field Theory, we want to apply the laws of quantum mechanics to the field itself, and not to particles. A field ...

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Let us work with the semi-disk $D$ with radial time flowing from the origin. Let us make the time $\tau=1$ be the contour boundary of the semi-disk while $\tau=0$ be the origin. Also, for simplicity let us neglect $b$ and $c$ for a while. An arbitrary first-quantized state of the string at $\tau=1$ is given by a functional $\Psi(X|_{\tau=1}(\sigma))$. A ...

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TL;DR: JoshuaTS' answer is exactly right: (Minus) the 3rd term ${\cal V}_3=\frac{1}{8\pi G}(\nabla\Phi)^2$ is the energy density of the gravitational field. In total OP's Lagrangian density contains 3 terms: ${\cal L}={\cal T}_1-{\cal V}_2-{\cal V}_3$. Here ${\cal T}_1$ is a kinetic term for matter, while ${\cal V}_2=\rho\Phi$ is an interaction/source term ...

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The last term represents the energy that is carried in the gravitational field itself. If you are familiar with electrostatics, this would be the equivalent of the statement that the energy stored in an electric field is $\frac{1}{2}\epsilon|\mathbf{E}|^2=\frac{1}{2}\epsilon(-\nabla V)^2$, where $V$ is the electric potential (the electric equivalent to $\phi$...

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The magnetic field originating from a bar magnet is continuously decreasing when moving away from the bar magnet. It does not have "lines" of stronger field strength. The lines that are forming are a consequence of the magnetic fields generated by the iron fillings themselves. An iron particle possesses a magnetic moment that aligns with the ...

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As far as I can see, your formula is correct. Let's transform this formula slightly. From $[\alpha_k,\alpha_{-k}] = 0$, it follows $$\varphi_k \equiv \frac{v_k}{u_k} = \frac{v_{-k}}{u_{-k}}.$$ Last equality together with $|u_k|^2 - |v_k|^2 = 1$ and $|u_{-k}|^2-|v_{-k}|^2 = 1$ leads to equalities $$u_k = \frac{e^{i\gamma_k}}{\sqrt{1-|\varphi_k|^2}}, \ v_k =... 2 Maxwell's equations describe a massless vector (spin-1) U1-gauge field (the photon-field). Other particles have different properties (spin, mass, coupling to other fields), and have different equations of motion. 1 Maxwell's equations can be written as a massless wave equation and this is a special case of Einstein's energy-momentum relation in wave form. The general case is the Klein-Gordon equation, which is satisfied by any free non-interacting field. 3 Equations of motion (EOM) are typically the equations that determine the time-evolution of the system. E.g. in Newtonian mechanics, Newton's 2nd law is the EOM. (One should avoid referring to the kinematic suvat-equations as EOM to avoid confusion.) For Lagrangian systems, the Euler-Lagrange (EL) equations are referred to as EOM, even in case of EL ... 0 The practical (although probably not the most rigorous) definition comes from looking at the derivatives in the equation. Equations of motion describe the evolution of the position of particles, so the independent variable is time (or a parameter like proper time) and the dependent variables are spatial position (or spacetime event locations). You'll see a ... 4 Field equations tell you how fields change in spacetime, whereas equations of motion tell you how arbitrary physical objects move in spacetime. In other words, field equations are equations of motion for a field. Normally the term EOM is used in classical mechanics to denote the motion of a single body or system of bodies (though it certainly is used in ... 8 There is no "precise distinction" between these terms. The "field equations" are just important equations of a field theory, which may or may not be the equations of motion for that theory. And even which equations are "equations of motion" is not unique! In the Lagrangian formulation of electromagnetism coupled to a charged ... 0 So, to complete the half solution in the answer one need only notice that multiplying the \frac{\partial S}{\partial\phi^i} the expression for F^{ij} in terms of F^{ia} and F^{ijk}, one is left with$$\frac{\partial S}{\partial\phi^i}F^{ij}=-\frac{\partial S}{\partial\phi^i}F^{ia}R^j_a.$$However, recall that this trivial gauge transformation was ... 0 I have done more direct approach to this task in first orfder in \epsilon. Let's start with dirct calculation of propogator:$$ \langle \phi(p) \phi(-p) \rangle = \frac{1}{p^2+m^2} - \frac{4 \cdot 3 \cdot g}{(p^2 + m^2)^2} \int^\Lambda \frac{d^{4-\epsilon}k}{(2\pi)^{4-\epsilon}} \frac{1}{k^2+m^2} + \dots  \int^\Lambda \frac{d^{d}k}{(2\pi)^d} \frac{...

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FWIW, Dirichlet boundary conditions (BCs) are not the only type of BCs. There are also e.g. Neumann or Robin BCs. The relevant BCs depends on the physical system at hand. This is true even if no variational formulation exists. In the context of a variational problem, say a stationary action formulation, if the Lagrangian depends on spacetime derivatives of ...

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3/2 spin particles are studied in the framework of Supergravity. So textbooks and lectures on Supergravity will certainly contain a study on 3/2-fermions since the supersymmetric partner(s) of the graviton, the gravitino(s) are 3/2-spin particles. On the other hand, as such particles are not part of the Standard Model (or variants of it) and Supergravity is ...

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If not, why not? Because we have not yet observed a fundamental particle with $3/2$ spin. If yes, how would one build such a theory? The Rarita–Schwinger equation.

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I think the local character comes from analyzing known phenomena. A widely studied theory like Electromagnetism has a gauge invariance in the classical formulation. We see therefore that changing gauge and having the same theory might persist going quantum, having it in the quantum regime assures us that it will persist classically. Then, we see that in ...

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Here is one approach: Let us for simplicity only consider static (i.e. time-independent) configurations. The time-dependent case is left to the reader. The stationary solutions are the 2 ground states, the kink and the antikink. (The 2 latter have a moduli parameter.) The 2 ground states are obviously locally stable. That the kink and the antikink are ...

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