Questions tagged [path-integral]

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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11
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1answer
3k views

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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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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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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Why not use the Lagrangian, instead of the Hamiltonian, in nonrelativistic QM?

Undergraduate classical mechanics introduces both Lagrangians and Hamiltonians, while undergrad quantum mechanics seems to only use the Hamiltonian. But particle physics, and more generally quantum ...
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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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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 pre-...
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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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Classical Limit of the Feynman Path Integral

I understand that in the limit that $\hbar$ goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation ...
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8answers
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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) = \|K(b,a)...
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Proof that the effective/proper action is the generating functional of one-particle-irreducible (1PI) correlation functions

In all text book and lecture notes that I have found, they write down the general statement \begin{equation} \frac{\delta^n\Gamma[\phi_{\rm cl}]}{\delta\phi_{\rm cl}(x_1)\ldots\delta\phi_{\rm cl}(x_n)}...
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In what sense is the proper/effective action $\Gamma[\phi_c]$ a quantum-corrected classical action $S[\phi]$?

There is a difference between the classical field $\phi(x)$ (which appears in the classical action $S[\phi]$) and the quantity $\phi_c$ defined as $$\phi_c(x)\equiv\langle 0|\hat{\phi}(x)|0\rangle_J$$ ...
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How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue?

I came across the S. Coleman's seminal papers 'Fate of the false vacuum' (http://dx.doi.org/10.1103/PhysRevD.15.2929, http://dx.doi.org/10.1103/PhysRevD.16.1762) where he describes the tunneling ...
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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 $$\tag{2-25} K(x_2 ,t_2;x_1, t_1)=\int_{x=x_1}^{x=x_2}\mathcal{D}x~ e^{\frac{i}{\hbar}S[2,1]}$$ ...
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Proof of Connected Diagrams

If $Z[J]$ is the generating functional for the path-integral, could any prove (or more reasonably, refer me to a proof) that $$W[J]\equiv\frac{\hbar}{i}\log\left(Z[J]\right)$$ "generates" only ...
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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 conditions)...
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1answer
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Why does the classical path of a particle give the dominant contribution in the path integral?

Why is it that the classical path of a particle gives the dominant contribution in the quantum mechanical path integral? How do we understand this?
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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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1answer
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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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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 $A_\mu\,^...
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How to prove useful property of logarithm of generating functional in QFT?

How to prove that $\ln(Z(J))$ generates only connected Feynman diagrams? I can't find the proof of this statement, and have only met its demonstrations for case of 2- and 4-point.
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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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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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Derivative interaction: $\mathcal{H}_\mathrm{int}\neq - \mathcal{L}_\mathrm{int}$. Question about Feynman Rules

As we known, if there is time derivative interaction in $\mathcal L_\mathrm{int}$, then $\mathcal{H}_\mathrm{int}\neq -\mathcal{L}_\mathrm{int}$. For example, Scalar QED, $$ \begin{aligned} \mathcal{...
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1answer
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What does the identity operator look like in Quantum Field Theory?

In texts on ordinary quantum mechanics the identity operators \begin{equation}\begin{aligned} I & = \int \operatorname{d}x\, |x\rangle\langle x| \\ & = \int \operatorname{d}p\, |p\...
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Are the path integral formalism and the operator formalism inequivalent?

Abstract The definition of the propagator $\Delta(x)$ in the path integral formalism (PI) is different from the definition in the operator formalism (OF). In general the definitions agree, but it is ...
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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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6answers
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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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2answers
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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 $\langle\mathcal{T}\phi(x_1)\phi(x_2)\rangle_{op}=\theta(x_1-x_2)\phi(x_1)\phi(x_2)+\theta(x_2-x_1)\phi(x_2)\...
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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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Gelfand-Yaglom theorem for functional determinants

What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form $Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$. ...
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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 (v1+v2)...
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1answer
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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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Equivalence of canonical quantization and path integral quantization

Consider the real scalar field $\phi(x,t)$ on 1+1 dimensional space-time with some action, for instance $$ S[\phi] = \frac{1}{4\pi\nu} \int dx\,dt\, (v(\partial_x \phi)^2 - \partial_x\phi\partial_t \...
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Problems while Wick rotating the path integral

I am trying to begin from the path integral of QM and write the Euclidean version of it performing the Wick rotation but it seems that I am missing a few things. For simplicity I work on 1 dimension ...
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2answers
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Gaussian integral with imaginary coefficients and Wick rotation

Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking ...
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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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Why are the Nambu-Goto action and Polyakov action equivalent at quantum level?

It's a well known elementary fact that the Nambu-Goto action $$S_{NG} = T \int d \tau d \sigma \sqrt{ (\partial_{\tau} X^{\mu})^2 (\partial_{\sigma} X^{\mu})^2 - (\partial_{\sigma} X^{\mu} \partial_{\...
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1answer
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What's the relation between path integral and Dyson series?

If one solves the Schrodinger equation $$i\hbar\partial_tU(t,0) = H U(t,0)$$ for time evolution operator $U(t,0)$, one can get the following Dyson series $$U(t,0) = \sum_n(\dfrac{-i}{\hbar})^n\...
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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 + V(x(\tau))\right]}...
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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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Is zero-point energy real?

In the context of canonical quantization, the ground state/zero-point energy of a harmonic oscillator is $$ E_0 = \frac{1}{2}\hbar \omega. $$ The vacuum is alleged to be permeated with this zero-point ...
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How exact is the analogy between statistical mechanics and quantum field theory?

Famously, the path integral of quantum field theory is related to the partition function of statistical mechanics via a Wick rotation and there is therefore a formal analogy between the two. I have a ...
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1answer
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Propagators, Green’s functions, path integrals and transition amplitudes in quantum mechanics and quantum field theory

I’m trying to make a simple conceptual map regarding the things in the title, and I'm finding that I’m a little perplexed about a couple of items. Let me summarize a few things I regard as being true, ...
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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 ...
18
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1answer
986 views

Faddeev-Popov Gauge-Fixing in Electromagnetism

Reading section 9.4 in Peskin, I am wondering about the following: The functional integral on $A_{\mu}$ diverges for pure-gauge configurations, because for those configurations, the action is zero. ...
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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 defect. ...
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1answer
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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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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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Can we do path integrals in gauge theories without fixing a gauge?

I am aware that when quantizing gauge theories with a path integral, one needs to add a gauge fixing term to avoid over-counting gauge related field configurations. From an aesthetic perspective, I ...
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Physical meaning of partition function in QFT

When we have the generating functional $Z$ for a scalar field \begin{equation} Z(J,J^{\dagger}) = \int{D\phi^{\dagger}D\phi \; \exp\left[{\int L+\phi^{\dagger}J(x)+J^{\dagger}(x)}\phi\right]}, \end{...