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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Amplitude $\langle0|e^{-iHT}|0\rangle$ in A. Zee's QFT In A Nutshell

In his Quantum Field Theory In a Nutshell, in page 12, (Second Ed), A Zee says that conventionally, the amplitude $\langle0|e^{-iHT}|0\rangle$ is denoted by $Z$. In the next paragraph, he considers ...
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143 views

Path integral in quantum mechanics

I am confused by the derivation in Srednicki QFT's chapter 6 from (6.8) to (6.9). In (6.8), we have $$<q'',t''|q',t'>~=~\int DqDp \exp[i\int_{t'}^{t''}dt(p\dot{q}-H(p,q))],\tag{6.8}$$ and ...
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1answer
67 views

Path integral measure Fourier transformation for case of real field

Let's have $$ Z[J] = \int D \varphi e^{iS[\varphi , J]}, $$ where $\varphi$ denotes real scalar field. Let's make Fourier transform, $$ \varphi (x) = \int e^{iqx}\varphi (q), \quad \varphi^{*} (q) = ...
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Deriving effective model without integrating out degrees of freedom in path integral formalism?

In path integral formalism of quantum field theory (particle physics or condensed matter), one can in principle integrate out part of the degrees of freedom so as to attain an effective model ...
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58 views

Does the path integral measure have dimension?

For example, in the field functional integral: $$\int D\phi \ e^{S[\phi]} $$ Does the $D\phi$ here have dimensions?
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What is the relation between phase space formulation with Wigner quasi-probability distributions and path integral formulation of quantum mechanics?

I am trying to conceptually connect the two formulations of quantum mechanics. The phase space formulation deals with Wigner quasi-probability distributions on the phase space and the path integral ...
16
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1answer
302 views

Quantum symmetries that are not classical symmetries

An anomaly is a symmetry of the classical action that fails to be a symmetry of the path integral, due to non-invariance of the path integral measure. Does it ever occur that the opposite thing ...
3
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1answer
194 views

Time-dependent Schrodinger equation from variational principle

In the paper, "Density-functional theory for time-dependent systems" Physical Review Letters 52 (12): 997 the authors mentioned that the action $$ A= \int_{t_0}^{t_1} dt \langle \Phi(t) | i ...
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1answer
193 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 ...
3
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1answer
201 views

Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
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0answers
37 views

Anomaly and Weyl spinors

I try to better understand anomalies in QFT and I've got a question concerning derivation of axial anomaly in Terning's lectures (page 12) Consider a theory of Weyl fermions coupled to a gauge field ...
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66 views

Path integrals and Weyl ordering in Peskin and Schroeder [duplicate]

On pages 280-281 of "An Introduction to Quantum Field Theory" by Peskin and Schroeder, the authors discuss the path integral formulation for a general quantum system and briefly mention Weyl ordering. ...
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50 views

How to calculate the expectation value of position/momentum using path integrals?

We have the formula: \begin{equation} \langle F \rangle = \frac{\int Dx \times F[\phi] exp\{i/\hbar S[\phi]\}}{\int Dx \times exp\{i/\hbar S[\phi]\}} \end{equation} Now, I am wondering how a change ...
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2answers
452 views

Quantum to classical mapping: quantum criticality and path integral Monte Carlo

I'm trying to understand the connections between quantum models in d dimensions and classical models in (d+1) dimensions within two, possibly related, contexts: (i) in path integral monte carlo, the ...
5
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1answer
239 views

Calculating $\mathrm{Tr}[\log \Delta_F]$

I am stuck with this problem for quite sometime. I have a propagator in the momentum representation (from this question), which looks like $$ \widetilde\Delta_F(p) = ...
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36 views

Integration measure

Consider the field being decomposed into a orthogonal and completed basis: $\Phi(x) = \sum_n c_n \phi_n(x)$ (or $\Phi(x) = \int dk c_k \phi_k (x)$, if continuous) The notation: $\phi_n(x) = ...
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50 views

Polology in Functional Integration

Completeness of Hilbert space (on-shell states) is a very powerful concept in canonical quantization, for example, to study the nonperturbative characteristics of the S-matrix, like polology (pole and ...
2
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1answer
78 views

Issues with the Operator to State map using Path Integral

Suppose your QFT has a Hilbert space $\mathcal{H}$, and let $\text{End}(\mathcal{H})$ be the set of operators on $\mathcal{H}$. It is often stated that in QFT there is a map $$\mathcal{F}: ...
2
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1answer
63 views

Length path integral

Let's consider a 2-dimensional Euclidean plane. The length between two points $a$ and $b$ can be defined in the following way: $$ (ab) := \inf_{\gamma} \,\int_0^1 d\tau \,\sqrt{\delta_{ab} ...
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1answer
79 views

Writing down many particle Hamiltonian

We are given that \begin{align}\mathrm{tr} e^{-\frac{i}{\hbar}\hat{H}t}&= \int D[a_1,\dots,a_n]\times\\&\qquad\exp\left[\int_0^t dt' \left(\frac{1}{2}\sum_j ...
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1answer
109 views

Perspectives of QFT [closed]

From the answer to this question Computing $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$, I have discovered that there is two perspectives to QFT. I am doing a course which is unfortunately a summary of QFT and ...
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60 views

Feynman Path integrals in space with holes in it [closed]

Feynman Path Integrals are a way of calculating the wave function of quantum mechanics. It usually integrates every possible path through all of space. I wonder if there is any study of Feynman path ...
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2answers
189 views

Why isn't the path integral rigorous?

I've recently been reading Path Integrals and Quantum Processes by Mark Swanson; it's an excellent and pedagogical introduction to the Path Integral formulation. He derives the path integral and shows ...
3
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1answer
55 views

Computing $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$

Given Hamiltonian $H=\frac{P^2}{2}+\frac{\omega^2}{2}Q^2$, compute $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$, where $T$ is the time-ordering of the product, $|0\rangle$ is the ground state. Now set ...
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Weinberg's spontaneous broken symmetries

Steven Weinberg in his second volume of QFT's book (in section about spontaneously broken symmetries, in subsection about Goldstone bosons) writes following: if we have linear transformation of ...
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53 views

How do we arrive on this kernel equation?

In Feynman and Hibbs, we see the following equation: $$K(b,a)~=~\sum_{\text{paths from $a$ to $b$}} \phi [x(t)] \tag{2-14}$$ which is valid always. Now, they write $$\phi[x(t)] ~=~ \text{const} ...
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limit as $x_1 \to x_0$, propagator for the harmonic oscillator

Consider a non-relativistic particle of mass $m$, moving along the $x$-axis in a potential $V(x) = m\omega^2x^2/2$. use path-integral methods to find the probability to find the particle between ...
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144 views

Getting Slavnov-Taylor identity

Let's have generating functional in path integral form for gauge $SU(n)$ theory with interaction: $$ \tag 1 Z[J] = \int DB D\bar{\Psi}D\Psi D\bar{c}Dc e^{iS}. $$ Here $$ S = S_{YM}(B, \partial B) + ...
3
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1answer
114 views

Tadpole diagrams in $\phi^3$ theory

In "Quantum Field Theory" by Mark Srednicki, Chapter 9 page 67, after he proves that $\langle 0|\phi(x)|0 \rangle$ vanishes (meaning sum of all connected diagrams with a single source is zero), he ...
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2answers
359 views

Feynman paths of FTL velocity have imaginary momentum?

In this answer it is discussed that Feynman path integrals sums amplitudes for all possible paths, including those that are not time-like. If you take the momentum-space path integrals, I would ...
3
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1answer
75 views

How is the functional integral over momentum performed in the case of the real scalar field?

Let's follow Peskin and Schroeder section 9.2, page 282. The Hamiltonian of a free real scalar field is $$H=\int{}d^3x[\frac{1}{2}\pi^2+\frac{1}{2}(\nabla\phi)^2+V(\phi)]$$ so the expression for ...
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1answer
32 views

Reducing unitary evolution operator of a two-spin system to the evolution operator of one of the spins

Consider a system of two spins $s_1$ and $s_2$, each of which can be in one of two states, represented by 0 or 1. A basis for the Hilbert space of this system would be {|0,0>,|0,1>,|1,0> and |1,1>}, ...
2
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1answer
98 views

Given a QFT Hamiltonian, is there a unique Lagrangian?

Consider a QFT in one spatial dimension specified by the following Hamiltonian density: $\mathcal{H} = -i \phi^\dagger \frac{\partial}{\partial x} \phi + V(\phi^\dagger,\phi)$ where $\phi$ is a ...
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2answers
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“Find the Lagrangian of the theory”

I've heard a few of my professors throw around the term "finding the Lagrangian of a theory". What exactly is this referring to. From what I understand it seems that you determine invariances ...
2
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2answers
136 views

If path integrals aren't well-defined, how can they have any physical meaning?

I am confused about a particular point about the nature of path integration. According to what I've read, what we really mean when we say functional integration is \begin{equation} ...
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9answers
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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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54 views

Feynman diagrams with classical apparatus on the perturbative region

on 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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1answer
51 views

Delta functional in path integrals - reference needed

In a few articles dealing with path integral quantization I came across some calculations where apparently identities of the form \begin{equation} \int (\mathcal{D}\Phi)\, ...
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38 views

Where can I find a video on Liouville Non-Critical String Path Integral?

Would anybody know of a video lecture course in which Polyakov's non-critical string path integral $$Z = \int D [\psi(\xi)]\, \, \exp \left\{ \frac{D-26}{48 ...
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32 views

Can someone explain how to do a functional integral through an example?

I want to learn how to do path integrals as used in quantum field theory. I understand the specific type of integral used in qft is called a functional integral. I looked at Wikipedia and checked in ...
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1answer
321 views

How do I solve this Gaussian path integral?

Suppose $$ Z = \int \mathcal D[\phi^*] \mathcal D[\phi] \exp(\phi^*A\phi + \phi B\phi) $$ where $A$ and $B$ are operators. I know how to solve a Gaussian path integral involving only $\phi^* A \phi$ ...
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5answers
679 views

Plants and Quantum Mechanics!

So, 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 ...
6
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1answer
132 views

Uses for Action from Lagrangian Mechanics

In my course on Lagrangian/Hamiltonian mechanics I noticed that we dealt with finding the stationary point of the change in action $ \delta S $ and we were never really doing anything with $ S $ ...
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1answer
419 views

Mathematically, what is the kernel in path integral?

Mathematically, what is the kernel in path integral? At first, I thought that it is the kernel in the integral transform because when we use the (physical) kernel to transform the wave function (Eq ...
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4answers
513 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 ...
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0+0 self interacting QFT - $ e^{-\sin^2 x}$ type integral — Bessel function expansion around infinity

In a physics paper (here) I found this variant of the Bessel function of the first kind. $$ \tag{1} Z(g) ~=~ \frac{1}{\sqrt{g}} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{-\frac{1}{2g} \sin^2 x} \, dx ...
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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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probability amplitude and path integrals [closed]

Recently, I have been learning about path integrals and I was wondering, can the probability of a certain path be weighted more in a path integral? Said in another way, can certain paths have more ...
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Are unitarity and locality properties of quantum field theory somewhat capsuled in these propierties of the action?

Feynman path integral weighs all paths by a factor $e^{i\frac{S}{\hbar}}$, where $S=\int \! L \, \mathrm{d^4}x$ Two questions: Is relatedthe fact that the argument of the exponential is imaginary ...
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51 views

Path integral formulation for an optimization quantum mechanics problem

I have been working on a quantum mechanics problem I asked here and someone recommended to use path integrals. I learned about path integrals but I couldn't find out how to finding the most optimized ...