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1

The propagator (kernel) can always be expressed as the path integral with a suitably chosen action: $$ K(b,a) = \int \mathcal{D}[\psi]\ e^{iS[\psi]/\hbar} \ , $$ where $\psi$ denotes the configuration of your system. The subtlety comes in choosing an appropriate $S$ for your situation. In single-particle, non relativistic quantum mechanics the ...


4

As $T = t_1 - t_0 \to 0$, we have$$\lim_{t_1 - t_0 \to 0} \langle x_1, t_1\,|\,x_0, t_0\rangle = \lim_{T \to 0}\left({{m\omega}\over{2\pi i\sin\omega T}}\right)^{1/2}\text{exp}\left[{{im\omega}\over{2\sin\omega T}}\left(\left(x_1^2 + x_0^2\right)\cos \omega T - 2x_0x_1\right)\right]$$$$=\lim_{\epsilon = iT/m \to 0} ...


1

Comments to the question (v1): Ref.1 is considering $\varphi^3$ theory $$\tag{1} {\cal L}(J)~=~\frac{1}{2}Z_{\varphi}\partial^{\mu}\varphi\partial_{\mu}\varphi - \frac{1}{2}Z_{m}m^2\varphi^2 - \frac{1}{6}Z_{g}g\varphi^3+(Y+J)\varphi.$$ To read the Feynman diagrams in Ref. 1, note that the source $J(x)$ is drawn as a black bullet $\bullet$, and the ...


1

Since your question is about special relativistic phenomena i think the best way to answer your question si in the context of quantum field theory, the propagator is defined, in the case of a scalar field theory,as: $$W(x-y)=\langle0|T[\phi(x)\phi(y)]|0\rangle$$ which can be put in relation to the path integral forumlation of QFT via the following equation: ...


3

That's not a handwaving and I think that this particular question is covered in practically every textbook containing path integrals. First of all you should note that we can integrate by $\pi$ not touching the second exponent, i.e. $$\langle\phi_b\vert e^{-iHT}\vert\phi_a\rangle =\int \mathcal{D}\phi e^{i\int_0^T d^4x\mathcal{L}}\int\mathcal{D}\pi ...


1

The time evolution of the two spins can be separated if they are independent, i.e. if they don't interact. Under this assumption the time operator splits in the tensor product $$U_1\otimes U_2=(U_1\otimes I)(I\otimes U_2)$$ and therefore it is clear how to define the time evolution for the single spin: for the $j$th particle one simply needs to take the ...


2

I) Before we get to quantization and path integrals there are problems already at the classical level. The Legendre transformation is not well-defined without knowledge of the CCR. For instance if the CCRs for the complex bosonic scalar $\hat{\phi}$ and $\hat{\phi}^{\dagger}$ is zero, this would mean that OP's Hamiltonian density ${\cal H}$ is a pure ...


0

The momentum being real has totally and absolutely 100% nothing whatsoever to do with whether a tangent to a curve is timelike, lightlike or spacelike. The energy-momentum vector points in the direction of the tangent to the worldline of the particle. The worldline is in a real four dimensional spacetime, so its tangent has 4 real components. Whether tha ...


2

Comments to the question (v2): "Find the Lagrangian of the theory" typically means that you are given the (classical) equations of motion (EOMs) of some physical system, and are supposed to find the action functional $S$ so that the EOMs are (parts of) the Euler-Lagrange (EL) equations for $S$. Note that an action principle/Lagrangian formulation does not ...


1

Usually the terms "Lagrangian" and "theory" can be considered the same. For a new theory, you have a new Lagrangian. For example, when we say "QED is different from QCD", we mean their Lagrangians are different. Each theory has its own Lagrangian. Although, observable quantities (and especially the equation of motion) is more important than the Lagrangian. ...


3

When I said we don't actually do path integrals, what I meant to say is that we can do some very specific path integrals and the way we do them is rather ad-hoc. In other owrds, it's highly nontrivial and not straightforward. To show this, I'll do a very general path integral for you. (I had most of this typed up already for another reason.) The vacuum ...


0

According to Faddeev and Slavnov (Gauge fields: Introduction to Quantum Theory), "...all those properties of the Feynman integral that are used in practice in the perturbation theory are derived directly from the definition of the quasi-Gaussian integral and can be rigorously established independent on the issue of existence of Feynman integral measure. ...



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