Path integral formulation (Due to Feynman) is a major formulation of Quantum Mechanics along with Matrix mechanics (Due to Heisenberg and Pauli), Wave Mechanics (Due to Schrodinger), and Variational Mechanics (Due to Dirac). DO NOT USE THIS TAG for line/contour integrals.

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Rigor in quantum field theory

Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
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Why path integral approach may suffer from operator ordering problem?

In Assa Auerbach's book (Ref. 1), he gave an argument saying that in the normal process of path integral, we lose information about ordering of operators by ignoring the discontinuous path. What did ...
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Why not using Lagrangian, instead of Hamiltonian, in non relativistic QM?

When we studied classical mechanics on the undergraduate level, on the level of Taylor, we covered Hamiltonian as well as Lagrangian mechanics. Now when we studied QM, on the level of Griffiths, we ...
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The meaning of imaginary time

What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in ...
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Normalizing Propagators (Path Integrals)

In the context of quantum mechanics via path integrals the normalization of the propagator as $$\left | \int d x K(x,t;x_0,t_0) \right |^2 ~=~ 1$$ is incorrect. But why? It gives the correct ...
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Path integral vs. measure on infinite dimensional space

Coming from a mathematical background, I'm trying to get a handle on the path integral formulation of quantum mechanics. According to Feynman, if you want to figure out the probability amplitude for ...
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When does $\hbar \rightarrow 0$ provide a valid transition from quantum to classcial mechanics? When and why does it fail?

Lets look at the transition amplitude $U(x_{b},x_{a})$ for a free particle between two points $x_{a}$ and $x_{b}$ in the Feynman path integral formulation $U(x_{b},x_{a}) = \int_{x_{a}}^{x_{b}} ...
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1answer
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What is a path integral? [closed]

I was reading about path integrals because someone told me about it in this question. I read some articles about path integrals but couldn't understand it. Can you please explain path integral for me? ...
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6answers
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Why is the contribution of a path in Feynmans path integral formalism $\sim e^{(i/\hbar)S[x(t)]}$

In Feynman's book "Quantum Mechanics and Path Integrals" Feynman states that the probability $P(b,a)$ to go from point $x_a$ at time $t_a$ to the point $x_b$ at the time $t_b$ is $P(b,a) = ...
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Time ordering and time derivative in path integral formalism and operator formalism

In operator formalism, for example a 2-point time-ordered Green's function is defined as ...
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3answers
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Once a quantum partition function is in path integral form, does it contain any operators?

Once a quantum partition function is in path integral form, does it contain any operators? I.e. The quantum partition function is $Z=tr(e^{-\beta H})$ where $H$ is an operator, the Hamiltonian of the ...
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Several stationary points of the action functional

In QFT the principle of stationary action states that we choose fields that will make the action stationary but what if the action has many stationary points (for a fixed choice of boundary ...
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Why is the functional integral of a functional derivative zero?

I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e. ...
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1answer
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In Path Integrals, lagrangian or hamiltonian are fundamental?

When studying path-integrals one question arose to my mind... Which presentation is more fundamental to calculate the propagator? The one based on the Hamiltonian (phase space)? $$K(B|A) = \int ...
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How does light know which path is fastest?

We know from Fermat's principle of least time that light follows the fastest path. But how does light know which path is the fastest?
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Normalization of the path integral

When one defines the path integral propagator, there is the need to normalize the propagator (since it would give you a probability density). There are two formulas which are used. 1) Original ...
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1answer
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More on the Feynman Path Integral Formula in Brian Cox' Lecture and its Consequences

This is a continuation of this question about Brian Cox' lecture Night with the Stars. I know the main steps to get from $K(q",q',T)=\sum_{paths}Ae^{iS(q",q',T)/h}$ to $\Delta t ...
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4answers
855 views

Quantum mechanics textbooks that use path integrals

I'm looking for a textbook in quantum mechanics that relies heavily on Green functions and the path integral formalism to supplement my QM books. I want to do some calculations using alternative ...
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1answer
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Classical limit of the path integral formulation of quantum mechanics

It is well-known that if $S \gg \hbar$, then the classical path dominates the Feynman path integral. But is there some to show that if $S\gg\hbar$, then the particle's trajectory will approach the ...
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1answer
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Feynman Path Integral Formula in Brian Cox' “A Night with the Stars” Lecture

The Youtube link keeps breaking, so here is a search on Youtube for Brian Cox' A Night with the Stars lecture. Pause the video on 40.32minutes. What you see he said is called Feynman's Path Integral. ...
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Equivalence Theorem of the S-Matrix

as far as I know the equivalence theorem states, that the S-matrix is invariant under reparametrization of the field, so to say if I have an action $S(\phi)$ the canonical change of variable $\phi \to ...
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On-shell symmetry from a path integral point of view

Normally supersymmetric quantum field theories have Lagrangians which are supersymmetric only on-shell, i.e. with the field equations imposed. In many cases this can be solved by introducing auxilary ...
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Wick rotation in field theory - rigorous justification?

What is the rigorous justification of Wick rotation in QFT? I'm aware that it is very useful when calculating loop integrals and one can very easily justify it there. However, I haven't seen a ...
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5answers
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What is the path integral exactly?

I asked a question here about path integrals and QFT. I just want to confirm something. Is the path integral in quantum field theory a mathematical tool only? I thought the path integral meant that ...
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5answers
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Plants and quantum mechanics!

I have been working on quantum biology and found something interesting that I would like to write an equation for. Scientists have wondered how plants have such a high efficiency in photosynthesis; ...
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1answer
282 views

Susy QM and Atiyah-Singer index theorem

Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and lagrangian $$ \mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k ...
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1answer
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Instantons and Non Perturbative Amplitudes in Gravity

In perturbative QFT in flat spacetime the perturbation expansion typically does not converge, and estimates of the large order behaviour of perturbative amplitudes reveals ambiguity of the ...
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Faddeev-Popov Ghosts

When quantizing Yang-Mills theory, we introduce the ghosts as a way to gauge-fix the path integral and make sure that we "count" only one contribution from each gauge-orbit of the gauge field ...
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1answer
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Differential equation (Greens function) satisfied by the kernel using path integrals

I'm reading Feynman and Hibbs, Quantum Mechanics and Path Integrals. How do I show that the kernel $$\tag{2-25} K(x_2 t_2;x_1 t_1)=\int e^{\frac{i}{\hbar}S[2,1]}\mathcal{D}x$$ satisfies the ...
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2answers
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In what sense is the path integral an independent formulation of Quantum Mechanics/Field Theory?

We are all familiar with the version of Quantum Mechanics based on state space, operators, Schrodinger equation etc. This allows us to successfully compute relevant physical quantities such as ...
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856 views

Physical Interpretation of the Integrand of the Feynman Path Integral

In quantum mechanics, we think of the Feynman Path Integral $\int{D[x] e^{\frac{i}{\hbar}S}}$ (where $S$ is the classical action) as a probability amplitude (propagator) for getting from $x_1$ to ...
6
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1answer
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Free Particle Path Integral Matsubara Frequency

I am trying to calculate $$Z = \int\limits_{\phi(\beta) = \phi(0) =0} D \phi\ e^{-\frac{1}{2} \int_0^{\beta} d\tau \dot{\phi}^2}$$ without transforming it to the Matsubara frequency space, I can ...
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1answer
216 views

What is the constraint on the Gauge Potential in the Covariant Gauges?

One of the most common gauges in QED computations are the $R_{\xi}$ gauges obtained by adding a term \begin{equation} -\frac{(\partial_\mu A^{\mu})^2}{2\xi} \end{equation} to the Lagrangian. ...
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4answers
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How is the path integral for light explained, or how does it arise?

In a Phys.SE question titled How are classical optics phenomena explained in QED (Snell's law)? Marek talked about the probability amplitude for photons of a given path. He said that it was ...
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1answer
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Boundary conditions in holomorphic path integral

Consider the holomorphic representation of the path integral (for a single degree of freedom): $$ U(a^{*}, a, t'', t') = \int e^{\alpha^{*}(t'') \alpha(t'')} \exp\left\{\intop_{t'}^{t''} dt \left( ...
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2answers
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Lagrangians combining terms with 1 and 2 derivatives

How are field theory Langrangians treated when some terms have 2 derivatives but others have only 1? Because the number of derivatives in a Lagrangian term is more easily even than odd, the ...
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Is every quantum measurement reducible to measurements of position and time?

I am currently studying Path Integrals and was unable to resolve the following problem. In the famous book Quantum Mechanics and Path Integrals, written by Feynman and Hibbs, it says (at the beginning ...
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1answer
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Subtlety of analytic continuation - Euclidean / Minkowski path integral

I subconsciously feel not fully comfortable about Wick rotating or analytic continuation from Euclidean to Minkowski space. I simply wonder whether there is any subtlety here, and when we need to be ...
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Semiclassical limit of Quantum Mechanics

I find myself often puzzled with the different definitions one gives to "semiclassical limits" in the context of quantum mechanics, in other words limits that eventually turn quantum mechanics into ...
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1answer
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Can nowadays spin be described using path integrals?

In Feynmans book, "Quantum mechanics and Path Integrals" he writes in the conclusions (chapter 12-10) With regards to quantum mechanics, path integrals suffer most grievously from a serious ...
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1answer
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Path integral with zero energy modes

Consider the field integral for the partition function of a free non-relativistic electron in a condensed matter setting, i.e. $$ Z = ∫D\bar\psi D\psi \exp\left(-\sum_{k,ω} \bar\psi_{k,ω} (-iω + ...
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What type of non-differentiable continuous paths contribute for the path integral in quantum mechanics

Consider the path integral for a 1D particle subjected to a potential $V(x)$ in imaginary time $$ \int_{x(0)=x_0}^{x(T)=x_T} [dx] \, e^{- \int_0^T d\tau \left[\frac{1}{2}\dot{x}^2 + ...
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The problem of a relativistic path integral

Many books have described the path integral for non-relativistic quantum. For example, how to get the Schrödinger equation from the path integral. But no one told us the relativistic version. In fact, ...
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4answers
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How can there be a quantum field theory that predicts all particle masses?

Say I have a theory with only one (energy) scale, e.g. one given by the fundamental constants $$\epsilon=\sqrt{\dfrac{\hbar c^5}{G}}.$$ In this case, where I can't compare to something else, is ...
6
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1answer
309 views

Casimir forces and its associated Feynman propagator

This is a continuation to my previous question, in which I began an attempt solve the Casimir Force problem using path integrals. As one of the answers there suggest I solve the Feynman propagator ...
5
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1answer
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Modern relevance of canonical quantisation [closed]

In some modern field theory texts such as Siegel's Fields it is claimed that canonical quantisation of fields is obsolete as it is not used it modern research papers. Thus, it should be removed from ...
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What is the value of a quantum field?

As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point ...
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Feynman diagrams with classical apparatus on the perturbative region

In QFT, one usually simplifies the interaction between fields and classical apparatus (sources, detectors, etc.) by assuming the classical devices only interact with the asymptotic on-shell states ...
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A quantum particle moving from A to B will take every possible path from A to B at the same time

If a quantum particle can take an unlimited number of paths to get from point A to point B wouldn't a quantum particle never get from point A to point B? A quantum particle takes every path at the ...
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Applications of the Feynman-Vernon Influence Functional

I am looking for a reference where the Feynman-Vernon influence functional was defined and used in the context of relativistic quantum field theory. This functional is one method to describe ...