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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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 $x_2$...
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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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102 views

Path integral with boundary and bulk terms

I was wondering if their is a general strategy for computing path integrals with a mix of boundary and bulk integral actions. Do people use divergence theorem to convert the action into bulk integrals,...
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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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Quantum field theory, particle interpretations and path integrals?

I am trying to find some names or models of a particle interpretation of quantum field theory which isn't a literal path integral approach? Are there any particle interpretations of quantum field ...
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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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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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How would a Lagrangian be used to recover the Schrodinger equation?

I heard that the Lagrangian is defined in the path integral formulation of quantum mechanics. How would the Lagrangian in this formulation be used to recover the Schrodinger equation that we normally ...
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Is the following a simpler viable alternative to Feynman's interpretation of the double slit experiment

Feynman suggested that there is an infinity of trajectories for a single electron travelling from the source to the phosphorescent screen. He said that one electron goes through both holes (Fig 4.10, ...
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path integrals: how/why can the phase be identified with the action?

In Peskin & Schroeder, chapter 9 introduces the functional methods. The idea, to recall, is simply to sum over all the possible paths: $U(x_a,x_b;T) = \sum_{\text{all paths}} e^{i . \text{...
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What's the role of classically forbidden paths in path integral?

I'm interested in how and how much classically-forbidden paths contribute to a path integral? Is there any good reference on the issue? Any discussion in QM or QFT context would be appreciated. EDIT:...
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Can path integrals be used to understand entanglement?

I like path integrals. I prefer to try to understand quantum phenomena in terms of path integrals rather than Hamiltonian mechanics. However, most of the standard texts on quantum mechanics start from ...
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Change of variables in path integrals

I need to evaluate a path integral which involves a set of fields $X=\left\{ \psi_i \right\}$: $$ I = \int \prod_i \mathcal{D} \psi_i e^{-S \left[ \left\{ \psi_i \right\} \right] } $$ In order to ...
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Intuition for Path Integrals and How to Evaluate Them

I'm just starting to come across path integrals in quantum field theory, and want to get the right intuition for the them from the start. The amplitude for propagation from $x_a$ to $x_b$ is typically ...
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Inclusion of information about external particles to calculate scattering amplitudes

In this (schematic) equation to calculate the scattering amplitude A by integrating over all possible world sheets and lifetimes of the bound states $$ A = \int\limits_{\rm{life time}} d\tau \int\...
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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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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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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 physics,...
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Is the Lagrangian density in field theory real?

As the Lagrangian in classical mechanics corresponds to energy, it must be real. But is that the case in quantum field theory? I mean, it should still correspond to some sort of energy, but what about ...
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Gaussian Integrals : Functional determinant expressed as a trace

Be $A_{ij}$ a symmetric matrix. Then I can easily write $$ \int \exp\left(-\frac{1}{2}\sum_{i,j}x_i A_{ij} x_j+\sum_{i} B_i x_i\right)\; d^nx= \sqrt{(2\pi)^n}\exp\left\{-\frac{1}{2}\mathrm{Tr}\log A\...
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Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
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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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What is the relationship between consistent histories and path integrals?

As can for example be learned from chapter I.2 of Anthony Zee's Quantum field theory in a nutshell, path integrals can be used to to calculate the amplitude for a system to transition from one state ...
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Path integral and geometric quantization

I was wondering how one obtains geometric quantization from a path integral. It's often assumed that something like this is possible, for example, when working with Chern-Simons theory, but rarely ...
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Chemical reaction as state transition?

When considering diffusion of chemicals, the reaction part is business of chemical kinetics, where the relevant characteristics of different substances come from collision theory together with some ...
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291 views

Importance of phase in probability amplitude in QFT

I have started teaching myself QFT from the textbook by A. Zee. From reading that book, my idea of a path integral in field theory is the probability amplitude to go from a given field configuration ...
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classical dynamics on group manifold SU(2)

I am trying to understand how to formulate classical dynamics on group manifold SU(2). This will be an exercise for me to the more advanced subject of path integral on group manifold. Does someone ...
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Calculating equation of motion using path integral

Suppose my action integral is $S=\int d^4x(\nabla \times A)^2$ and $\delta S$ gives $\delta S =\int d^4x [2(\nabla \times A).(\nabla \times \delta A)]$ I would like to calculate the coefficient of $\...
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Free Particle Propagator Using Path Integrals

I'm trying to recreate some work that a professor explained to me in his office, specifically deriving the free particle propagator going from $(y,0)$ to $(x,T)$ using the Feynman Path Integral. I'm ...
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Surface terms for field path integrals?

My question relates to something that I´ve seen in many books and appears in all its glory here: Ryder, pg 198 My question is about eq. 6.74. Which I repeat below: $$i \int {\cal D}\phi \frac{\delta ...
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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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411 views

Why can't the functional integral be derived in a mathematically rigorous way?

Why can't the functional integral be derived in a mathematically rigorous way? What are the obstacles that we have to overcome in order to achieve that goal?
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Relation between Dirac's generalized Hamiltonian dynamics method and path integral method to deal with constraints

What is the relation between path integral methods for dealing with constraints (constrained Hamiltonian dynamics involving non-singular Lagrangian) and Dirac's method of dealing with such systems (...
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Does the vacuum energy problem of quantum field theory only occur in the Hamiltonian approach, or also in the path integral approach and in AQFT?

In a standard QFT class, you're being indoctrinated that there is the "infinite vacuum energy density problem". (This is sometimes paraphrased as the "cosmological constant problem", which is in my ...
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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 ...
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Discrete version of Feynman path integrals

I've decided to put a very limited amount of my time into understanding the path integral formulation of quantum mechanics. I'm interested in the mathematical formalism more than the physics, so I'd ...
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398 views

Integrating over a gauge field in the field integral formalism

I'm currently trying to study a chapter in Altland & Simons, "Condensed Matter Field Theory" (2nd edition) and I'm stuck at the end of section 9.5.2, page 579. Given the euclidean Chern-Simons ...
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Feynman's sum over histories?

The concept requires all possible path's to be mapped out, and any remaining paths not cancelled out represent the most probable path of the object. Considering this: i) If "infinite" paths are ...
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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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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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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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Good resources for learning/reviewing complex time propagator formalism

I studied this at the beginning of my graduate degree but have to review it for my graduate exam. If it's not clear I'm talking about the $\beta = \frac{it}{\hbar} $ turning the integral of your ...
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Does path integral and loop integral in a Feynman diagram violate special relativity?

Consider a correlation function between two points $A(x_1,t_1)$ and $B(x_2,t_2)$, we need to integrate over paths which could be infinite long. But the time length $(t_1-t_2)$ is finite, so if $A$ and ...
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Which limit for matsubara frequency sum?

in the context of a simple toy problem for Feynman path integrals, I consider a two-site Hubbard model for spinless fermions. I expand the path integral to first order in the interaction $V$, which ...
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Results for the path integral formalism for a system with known start and end configuration?

The path integral provides a method for computing a time evolution by a weighted summing up all possible deviations. Is there such a method for a system, where one not only knows the initial ...
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Vacuum Wavefunctional

I am having this problem in understanding the vacuum wavefunctional in QFT. Hence this naive question. I mean, if someone say vacuum wavefunctional, I can think of an element like wavefunction as in ...
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Path integral formulation of quantum mechanics

I'm a mathematics student with not much background in physics. I'm interested in learning about the path integral formulation of quantum mechanics. Can anyone suggest me some books on this topic with ...
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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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Are there rigorous constructions of the path integral for lattice QFT on an infinite lattice?

Lattice QFT on a finite lattice* is a completely well defined mathematical object. This is because the path integral is an ordinary finite-dimensional integral. However, if the lattice is infinite, ...
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When can I use Wick's theorem?

Wick's theorem means that for fermions, a four point correlation function (for example) can be written in terms of two point correlation functions: \begin{equation} \langle b_l^\dagger b_l b_m^\...