Questions tagged [action]

The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.

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Calculation of gravitational Euclidean action of Schwartzchild BH

I am reading the paper of Gibbons and Hawking Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 where they compute the gravitational action of black holes. In ...
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Einbein in action for relativistic particles [duplicate]

Studying path integrals, in class we were looking for actions of relativistic particle. One of the possible action was $$S[x^\mu(\tau)]=-m \int d\tau\sqrt{-\dot{x}^\mu\dot{x}_\mu}$$ We introduced ...
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How is the Weyl lagrangian zero for Weyl spinors?

The action from which Weyl equation can be derived is $$S=i\int{\psi_{L/R}^\dagger\bar{\sigma}^\mu\partial_\mu\psi_{L/R}}$$ where $\bar{\sigma}^\mu=(1,\vec{\sigma})$. Imposing $\delta S=0$ we arrive ...
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1-Loop Approximation for Fermionic Effective Action

Given a partition function $$ Z = \int \mathscr{D}(\bar{\psi}, \psi) \, \mathrm{exp} \left( \, - S[\bar{\psi}, \psi] \right) $$ with fermionic (Grassmann) fields. I seek to calculate the effective ...
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Wedge product (Exterior product) in Palatini action

Excluding the factor, we can write the tetradic Palatini action and Holst action as respectively $$\epsilon_{IJKL}e^I\wedge e^J\wedge F^{KL}$$ $$e^I\wedge e^J\wedge F_{IJ}$$ where $e^I=e^I_\alpha \...
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Equality modulo equations of motion [closed]

What does Qmechanic mean by “equality modulo equations of motion” when talking about Lagrangian formulation/formalism and so on?
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Application of the Cartan Structure Equations seems to imply the Einstein-Palatini action is zero?

The Einstein-Palatini action can be written as $$ S = M_{pl}^2\int\varepsilon_{abcd}\left(e^a\wedge e^b\wedge R^{cd}\right), $$ where $e^a={e^a}_\mu\text{dx}^\mu$ is the basis one-form and $R^{ab}=\...
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First order quantum string action

Considering this post: Quantum String action the action given is of the lowest order but the effective action, for low energies, is given by: $$ S_{ef.}= -\frac{1}{2k^2} \left( S^{(0)}+ \alpha S^{(1)} ...
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How to calculate functional derivative correctly?

Let $\phi$ be a real scalar field and $J$ an arbitrary source function. Consider $$S_{E}[\phi, J]=\int d^{4} x\left[\frac{1}{2}(\partial_{\mu} \phi)(\partial^{\mu}\phi)+\frac{1}{2} m^{2} \phi^{2}+V(\...
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Obtaining General Relativity

Considering the action given in the following post:Induced gravitation action under what conditions does GR originate from this induced gravitation model? Also, what can we say about the cosmological ...
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Converting between Lagrangian and Hamiltonian formalisms of relativistic action

Hello everyone I am currently studying the relativistic free particle. I know that the action obtained in the Lagrangian formalism is $S = -m \int_{\tau_i}^{\tau_f} \sqrt{-\eta_{\mu \nu} \frac{dx^{\mu}...
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The action of the Kaluza–Klein reduction (Chapter 4 of “D-branes (Clifford Johnson)”)

In the first part of Section 4, the author gives \begin{equation} S = \frac1{16\pi G^N_{(5)}}\int(-G_{(5)})^{1/2}R^{(5)}d^5x = \frac1{16\pi G^N_{(4)}}\int(-G_{(4)})^{1/2}\Big(R^{(4)} - \frac32\...
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Clearing up some simple details of the types of symmetries involved in Noether's theorem

I would just like to ensure that I have fully understood the content of Noether's theorem and a few of its details. The generic statement of Noether's theorem is relatively straight forward however ...
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Some confusion regarding the specifics of the geometric formulation of Lagrangian mechanics and Noether's theorem

I would like to resolve a few problems I'm having regarding the exact procedure of Lagrangian mechanics when formulated as the tangent bundle of configuration space. These problems are not overly ...
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Is there an accepted Lagrangian for the transport equation?

Perhaps because it is so simple, I have not seen a lagrangian form of the transport equation $$(\partial_t + a \partial_x)q = 0.$$ This equation is first order, which makes obtaining it from the Euler-...
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Solving the action without solving for fields?

This is more of a philosophical question for a problem I'm trying to solve: are there examples in physics for which we evaluate the on-shell action without solving for the fields in the action? The ...
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Can we have conservation of momentum without conservation of energy?

According to Noether's theorem, if the Hamiltonian is invariant under translations in a given direction, then the corresponding linear momentum is conserved. And if the Hamiltonian is time-independent,...
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Constant term in action term in general relativity?

Question So I recently pondered the following. Let's say I have an $2$ actions $S_1$ and $S_2$ which differ by a constant: $$ S_1(\dot x_i, x_i) = S_2(\dot x_i, x_i) + \tilde c$$ Now their equations ...
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Integrating out bosons/fermions when the action is quadratic

Arovas and Auerbach, in their paper titled "Functional integral theories of low-dimensional quantum Heisenberg models" try to compute the free energy of $SU(N)$ models with the large $N$ ...
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Extra term in E.O.M. for Nambu-Goto action with general targetspace metric

Strangely I haven't found a derivation of the E.O.M. of a string in an arbitrary background metric $G_{\mu \nu}$ from the Nambu-Goto action. (Many places present it for the Polyakov action.) Upon ...
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Is there some alternatives to least action principle

The principle of least action seems to be one of the most fundamental of high-energy/fundamental interactions physics. But is there some other possibility ton construct a theory of interactions? Or, ...
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Why does this method of deriving the classical free particle Lagrangian not work?

I was reading volume two in Landau and Lifshitz's Course of Theoretical Physics (The Classical Theory of Fields). In it, Dr. Landau develops the relativistic Lagrangian as follows: one has $$S=\alpha\...
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Solving the shape of a vortex as an extermal problem

A friend told me the following riddle: Suppose you have shake a bottle around, causing water to spin with fixed angular speed. What is the shape of the vortex? The answer is a parabola, and one can ...
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In path-integral, when do we have to insert fact $i$ in front of the action $S$ in the exponent?

I have got stuck in these concepts for a fews days: Wick rotation, Euclidean spacetime and QED in gravity. Generally, in Minkowski space time, there is a factor $i$ in front of the action $S$, e.g., ...
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Looking for a Basic ( or “for Dummies” ) Explanation of the Lagrangian - Hamiltonian Relationship. ( Mathematician ) [duplicate]

(Mathematician here - first time physics.stack poster). I'm basically looking for as simple as possible explanation of the Hamiltonian - Lagrangian relationship. $\textbf{My understanding :}$ $\textbf{...
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How to derive the ADM form of this relativistic Hamiltonian?

The action for a general relativistic particle should be: $$S=\int d\tau\int d^3x \sqrt{-g} g^{\mu\nu}\dfrac{dx^\mu}{d\tau} \dfrac{dx^\nu}{d\tau} \,\delta^3(x-y)$$ where $g_{\mu\nu}$ is the 4-metric. ...
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Vanishing action integral for Gravitation Field

For a Gravitation Field Action Integral looks like: \begin{equation}\label{1} S_{gravity} = \frac{c^3}{16\pi G}\int R\sqrt{-g} d^4x. \end{equation} А Least Action Principle says the $\delta S_{...
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Application of Lagrange Multipliers in action principle

In Goldstein's Classical Mechanics, he suggests the use of Lagrange Multipliers to introduce certain types of non-holonomic and holonomic contraints into our action. The method he suggests is to ...
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Interest of having 2 system of coordinates in the four-dimensional volume form

In the following expression of Lagrangian in General Relativity : $$S=\int d^{4} x \sqrt{-g}\left(\frac{R}{16 \pi G}+\mathcal{L}_{\mathrm{M}}\right)$$ I understand that we can write for example : $$c\,...
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Convert a quasi-symmetry of the action into a strict symmetry

A quasi-symmetry of an action $S$ is a transformation of the fields that leaves the action invariant up to a boundary term (see e.g. the answer to this question). In contrast, let us call a ...
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Understanding the metric transformation under infinitesimal diffeomorphism

In my general relativity course, we are discussing infinitesimal diffeomorphisms defined by $x^{\mu}\rightarrow y^{\mu}(x) = x^{\mu} + \xi^{\mu}(x)$. We have been examining how different objects ...
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Born-Infeld term for $D$-brane low energy effective action

I am trying to reproduce the Born-Infeld term for the $D$-brane action as explained in Szabo's BUSSTEP Lectures and must admit I am utterly confused by some of the steps. This is a long and technical ...
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Relativistic mechanics: Does the action rely on a particular choice of the inertial frame?

We know that the Lagrangian of a relativistic particle is as follows: $$L = -mc^2\sqrt{1-(v/c)^2},$$ with the action being the integral of this Lagrangian with respect to time in the reference frame ...
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Lagrangian of a free particle in Special Relativity and equivalence between mass and energy

I am a bit confused on the way Landau derives the Lagrangian of the free particle in SR (L. Landau, E. Lifshitz - The Classical Theory of Fields) and his conclusions about the equivalence between mass ...
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Understanding the four-dimensional volume form in Action of Lagrangian

Into the following part below, I don't understand what is precisely a "four-dimensional volume form" implied in the integral below: For comparison, the ...
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Operator norm and Action

We define the norm of the operator as $\left\lVert A \right\rVert = \sup \frac{\left\lVert A\psi \right\rVert}{ \left\lVert A \right\rVert} = \sup \left\lVert A\psi \right\rVert$ for $A ∈ L(H)$. It is ...
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Why must the action be minimized? [duplicate]

In mechanics, the only physical route a particle can take is the one where action is minimized. Why is this true? Is there a proof?
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Lagrangian in reduced Horndeski Theory for i=2

I am trying to understand the calculations of the latest Charles Dalang's paper "Scalar and Tensor Gravitational Waves", arXiv:2009.11827. Since I just learned basic general relativity, I ...
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Is there any restriction on the Lagrangian of a system?

I have learned the calculus of variations in my previous semester, and now we are studying classical mechanics. What I found is that there is lots of lack of rigor in Lagrangian mechanics in ...
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The action in classical field theory

In calculating the action in classical field theory, why do we integrate over all of spacetime, thus over all of time, while we don't have to do that in ordinary particle action?
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Understanding how to derive the discrete Euler Lagrange equations

I learned how to derive Euler-Lagrange equations in classical lagrangian field theory: we start off a Lagrange density (which only depends on the fields themselves and their first order derivatives), ...
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What can “action” mean in “All the action takes place inside the rocket”?

This answer to "How fast is fuel escaping a rocket for it to reach escape velocity 11km/s?" includes the following: From Rocket and Spacecraft Propulsion: Principles, Practice and New ...
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When do two classically equivalent actions give the same quantum theory?

When we studies the beginnings of string theory, we looked at einbeins, and that because they were equivalent to the action $\int \sqrt{\eta_{\mu\nu}\dot{x}^\nu\dot{x}^\nu}$ then the einbein action ...
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Effective action quantum field

I am reading Peskin and Schroeder Section 11.4. They derive a formula for the effective action p.372 Equation 11.63 using a scalar field interaction, $$ \Gamma \left ( \phi _{cl} \right )=\int d^{4}...
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Action at a distance in Quantum Field Theory

Definitely, I don't mean entanglement here: Suppose we have an electron and proton situating some distance apart, there is an electrostatic force between them, and this force is mediated by virtual ...
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What’s the motivation for using Ricci scalar in action term for spacetime?

Why do we use Ricci scalar in the action equation for the spacetime? Why don’t we use other functions? Is it just intuition? What forces us to use that?
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What is the role of time (time interval) in principle of least action?

Action is represented by $S[Q(t)]$ where $Q(t)$ is the name of a single complete path in the configuration space of a system. The path starts at the point $q_i$ and ends at the point $q_f$. Suppose ...
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Equation of Motion from Action for a Scalar Field + Matter

In a review on quintessence, the equations of motion (EoM) for the action $$ S=\int\!\mathrm{d}^4x\sqrt{-g}\left(\frac{M_p^2R}{2}-\frac{g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi }{2}-V\left(\phi\...
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From Reissner-Nordstrom to Dilatonic Gravity and Einstein Equation

I am doing some calculations of this paper: https://arxiv.org/abs/1711.08482 and in particular I am having trouble with 2 parts: Dimensional Reduction (16): I have managed to get equation (16) from ...
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Lagrangian of free particle relativistic case

Why must the covariant Lagrangian of a free particle be a first-order differential?

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