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There are two distinct notions of symmetry in quantum theory which does include QFT. The former regards the structure of the space of the states (or also the algebra of observables). The latter uses the former notion but concerns the dynamical evolution. Wigner theorem regards the first type of symmetry. A symmetry (of first type), is a map $S$ from the ...


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Rather than asking why we should impose this invariance, I think it would make more sense to ask why we should relax it. The Lagrangian is a relativistic scalar, which means that it has to be invariant under any coordinate transformation. An infinitesimal change of coordinates is just one type of coordinate transformation. The physical interpretation is ...


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A central force is simply a force that is always directed towards a fixed point in space. Gravity can be treated as a central force in certain circumstances. The gravitational force acting on object $A$ orbiting around object $B$ can be approximated by a central force acting towards the centre of $B$ if $B$ is much more massive than $A$. By Newton's third ...


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No, homogeneity does not implies that the expansion is uniform. Homogeneous expansion could be anisotropic which would lead to different changes in length depending on orientation. A simple example to demonstrate this is the Kasner metric which is homogeneous but anisotropic. For a $(3+1)$ spacetime this metric could be written in the following form: $$ ds^...


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In general the tensor product of two irreducible representations is reducible. The best example is the coupling of two spin-1/2 states, which give $$ \frac{1}{2}\otimes \frac{1}{2}=0\oplus 1\, . $$ If one of the representation is 1-dimensional, then the result will usually remain irreducible. For instance, the alternating representation ${\cal A}$ of $S_n$...


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Liouville theory possesses conformal invariance on the quantum level, but its classical action isn't conformally invariant (because it contains dimensionful constants such as $\lambda'$).


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There are at least 2 issues with OP's discussion (v2): One should properly distinguish between total and explicit spacetime derivatives, cf. e.g. my Phys.SE answer here. In particular, an infinitesimal quasisymmetry of the Lagrangian (density), means by definition that the infinitesimal variation is a total (space)time divergence. Note that not all terms ...


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This is kind of like the analogue of kinetic energy in Newtonian physics. If you add in potential energy as well, then that can have an arbitrary zero.


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This is easiest done not using index notation. Define the column vector $$ \vec{\phi} = \begin{pmatrix} \phi_1 \\ \phi_2 \\ \phi_3 \end{pmatrix} $$ The derivative acts on the column vector as $\mathbf{1}_3\partial_\mu$ where $\mathbf{1}_3$ is the identity matrix on the same vector space in which $\vec{\phi}$ lives. That is, $$ \partial_\mu \vec{\phi} = \...


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Local (over scales like the Solar system at least) timelike Killing vectors do exist, otherwise we could not formulate and experimentally confirm any conservation low for energy. What can be said is that the observed expansion of the spatial sections of the universe does not permit a large scale timelike Killing vector orthogonal to those spatial ...


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You are nearly finished. Consider your result $$L' = L+ \theta(2xy(a-b)-3cxy^2) + O(\theta^2).$$ In order to make $$\theta(2xy(a−b)−3cx^2) = 0 \quad \text{for all } x \text{ and } y$$ you must have the conditions $a=b$ and $c=0$.


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I would say a way to understand why symmetries and scalar product are related is the following. Physical quantum (pure) states are not really vector in a Hilbert space, rather they are rays, that is equivalence classes of vectors which differ by a complex number (or normalized vectors which differ by a phase). Now the natural distance between quantum ...


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This answer is about symmetry in quantum systems in general in the context of Wigner's theorem; however, it is applicable to the case of quantum field theory. I'll consider, as a motivation, the case of classical systems which has analogous definitions of symmetry. A classical Hamiltonian system can be defined by the principle of least action, which in the ...


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I don't think it does imply the Lorenz gauge. You can take advantage fof the gauge freedoms to specify any scalar relationship between the magnetic vector and scalar electric potentials, which is equivalent to choosing a form for the function $\Psi$. For instance, the Lorenz gauge implies that $\Psi$ is any function that satisfies $$\nabla^2 \Psi = \frac{1}{...


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Maximally symmetric space is a space that is both homogeneous and isotropic. Such a space possesses the largest possible number of Killing vectors which in an n-dimensional manifold equals $n(n+1)/2$. The following holds for a maximally symmetric space: The scalar curvature $R$ is a constant. The Ricci tensor is proportional to the metric tensor, i.e., $R_{\...


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