The path-integral tag has no wiki summary.
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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 ...
14
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1answer
63 views
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 ...
14
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4answers
234 views
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 ...
13
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2answers
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Calculating correlation functions of exponentials of fields
In their book Condensed Matter Field Theory, Altland and Simons often use the following formula for calculating thermal expectation values of exponentials of a real field $\theta$:
$$ \langle ...
13
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2answers
161 views
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 ...
12
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1answer
214 views
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 ...
10
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1answer
1k views
The Concepts of Path Integral in Quantitative Finance
I realize that path integral techniques can be applied to quantitative finance such as option value calculation. But I don't quite understand how this is done.
Is it possible to explain this to me ...
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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 ...
9
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3answers
199 views
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 ...
9
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446 views
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 ...
8
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5answers
503 views
Why is the contribution of a path in Feynmans path integral formalism $\sim e^{(i/\hbar)S[x(t)]}$
In Feynmans 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) = ...
8
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2answers
136 views
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, ...
8
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1answer
482 views
Questions about the Dyson equation
I'm studying finite temperature many-body perturbation theory, and am trying to understand The Dyson equation. In particular, I'm using Mattuck - A guide to Feynman diagrams in the many body problem.
...
7
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3answers
852 views
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 ...
7
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4answers
380 views
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 ...
7
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1answer
538 views
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
I have come across a limit of Gaussian integrals in the literature and am wondering if this is a well known result.
The background for this problem comes from the composition of Brownian motion and ...
7
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1answer
250 views
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 ...
7
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3answers
269 views
How does the quantum path integral relate to the quantization of energy?
So, the quantum path integral is a generalization of the classical principle of least action- but here we know that all paths contribute something finite to the probability density. What confuses me ...
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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}} ...
6
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1answer
232 views
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 ...
6
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1answer
185 views
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] } $$
In order to ...
6
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4answers
344 views
Quantum mechanics textbooks that use path integrals
I'm looking for a textbook in quantum mechanics that relies heavily on Green functions and the path integral formalism to supplement my QM books. I want to do some calculations using alternative ...
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718 views
Wheeler-Feynman theory, QED without fields, vacuum polarization
Initially Wheeler and Feynman postulated that, the electromagnetic field is just a set of bookkeeping variables required in a Hamiltonian description. This is very neat because makes the point of ...
6
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1answer
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Differential equation (Greens function) satisfied by the kernel using path integrals
I'm reading Feynman and Hibbs, Quantum Mechanics and Path Integrals. How do I show that the kernel
$$\tag{2-25} K(x_2 t_2;x_1 t_1)=\int e^{\frac{i}{\hbar}S[2,1]}\mathcal{D}x$$
satisfies the ...
6
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2answers
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Surface terms for field path integrals?
My question relates to something that I´ve seen in many books and appears in all its glory here: Ryder, pg 198
My question is about eq. 6.74. Which I repeat below:
$$i \int {\cal D}\phi \frac{\delta ...
6
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2answers
308 views
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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4answers
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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 ...
5
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5answers
346 views
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 ...
5
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1answer
361 views
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 ...
5
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1answer
170 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 ...
5
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1answer
499 views
What is the meaning of the Fourier transform of Feynman propagator?
I know $K(a,b,t)$ is the probability amplitude of find a particle that starts at point a in b in a time t later. There is also an expression that sometimes is called green function:
...
5
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1answer
208 views
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.
$$ Z = ∫D\bar\psi D\psi \exp\left(-\sum_{k,ω} \bar\psi_{k,ω} (-iω + ...
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Calculating the the kernel using path integrals for quadratic lagrangians
I am reading Feynman and Hibbs on Path Integrals. In section 3.5, they show that the kernel for a lagrangian of the form $L=a(t)\dot{x}^2+b(t)\dot{x}x+c(t)x^2+d(t)\dot{x}+e(t)x+f(t)$ is ...
5
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279 views
Two paths having the same phase in the path integral approach
In the path integral approach to Quantum Mechanics, can two distinctly different paths of the possible infinite paths have the same phase, i.e can there be a bimodal distribution of the phases ...
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Gaussian Integrals : Functional determinant expressed as a trace
Be $A_{ij}$ a symmetric matrix. Then I can easily write
$$
\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 ...
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1answer
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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 ...
4
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1answer
547 views
Do derivatives anticommute with Grassmann variables and complex numbers in a many-body path integral?
I'm trying to learn how to do a many-body path integral for both fermions and bosons, and I'm stuck. I'm following Altland and Simons - Condensed Matter Field Theory, chapter 4. On page 167, equation ...
4
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315 views
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 ...
4
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1answer
214 views
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 ...
4
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1answer
299 views
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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1answer
223 views
Have the correlation functions of the XY spin chain model been calculated using a functional partition function with source terms?
Have the correlation functions of the XY spin chain model,
\begin{equation}
H=-\sum_l (J_x \sigma_l^x \sigma_{l+1}^x+J_x \sigma_l^y \sigma_{l+1}^y)-B\sum_l \sigma_l^z
\end{equation}
been calculated ...
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0answers
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When can the source term of a partition function be put in?
More specifically, in quantum field theory books, we usually have this:
\begin{equation}
Z = \int D(\bar{\psi}, \psi) e^{-S + \int_0^\beta d\tau \sum_l [\bar{\eta}_l (\tau) \psi_l (\tau) + ...
4
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4answers
382 views
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.
...
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2answers
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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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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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3answers
289 views
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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2answers
336 views
When is many-body perturbation theory valid?
I'm calculating expectation values (thermal, time-independent) using many-body perturbation theory, but I'm unsure how to work out what values the parameter I'm expanding the perturbation series in ...
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2answers
316 views
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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0answers
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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 ...
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2answers
236 views
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 ...
