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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26 views

How can a boundary have an equation of motion? (Quadratic gravity with boundaries)

This question is the following of my previous one on the appearance of quadratic gravity from a simple scalar field treated non-perturbatively. In the following, I will not specify the $\epsilon$ term ...
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63 views

Noether identities and the relativistic point particle

I am trying to better understand Noether identities, i.e. relations between equations of motion in the presence of gauge symmetries for the example of the relativistic point particle. Formally, a ...
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48 views

How can the scaling dimension of a scalar field be $1$ in a $1d$ CFT?

In a one-dimensional scalar field theory, the kinetic term of the action takes the following form: $$S_\text{kin} = \int_\mathbb{R} dt\, \frac{1}{2} \dot{\varphi}(t) \dot{\varphi}(t)\,, \tag{1}$$ with ...
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Is Weyl gauge together with Coulomb gauge a possible choice?

I am working in the Weyl gauge with an action in euclidean signature (so the Lagrangian is a Hamiltonian): \begin{equation} S=-\frac{1}{96\pi^2}\int_{\mathbb{R}^4} d^4 x\,\text{Tr}\left( E_i \cdot ...
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How can I show that the action of a SHO is a saddle-point solution if $t_{f} - t_{0} > T/2$?

In this post and in this post, QMechanic claims that a simple harmonic oscillator with Dirichlet boundary conditions has saddle-point solutions if $t_{f} - t_{0} > T/2$ where $T$ is the ...
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In a theory with spinor fields, what general condition on the action ensures that the stress-energy tensor can be made symmetric?

In general relativity, the stress-energy tensor is normally defined by $$ T^{ab}\equiv \frac{2}{\sqrt{|g|}}\,\frac{\delta S_m}{\delta g_{ab}}, \tag{1} $$ where $S_m$ is the "matter" action (...
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Boundary terms of action in AdS

Let's consider the action of a free scalar field in a space-time with metric $g_{\mu \nu}$ $$ S = -\frac{\eta}{2} \int d^{d+1}x \, \sqrt{g} \{g^{AB} \partial_{A} \phi \partial_{B} \phi + \frac{1}{2} m^...
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Help solving one dimensional third order Euler-Lagrange condition

I have the following Lagrangian: $$ \mathcal{L}(t, x, \dot{x}) = \frac{df}{dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} \dot{x} + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} b(...
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What is the reason behind the stationarity of action? [duplicate]

I am reading Goldstein right now to understand the least action principle. I understood that the action needs to be stationary under small variation and this specifies the equation of motion, but do ...
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96 views

Wald's approach to deriving the Einstein field equations and the Levi-Civita connection through Palatini's action

I'm reading Appendix E of Wald's General Relativity book and I'm a bit confused in how he derives the Einstein field equations and the Levi-Civita connection through Palatini's action. The Palatini ...
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Missing minus sign when taking derivative

I'm trying to understand to get the following formula (first formula on pg 33) in Altland Simons second edition: $$\Delta S \simeq \int d^m x (1 + \partial_{x_\mu} \, (\omega_a \, \partial_{\omega_a} \...
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43 views

Einstein-Hilbert action and antisymmetry of derivative of $g_{\mu \nu}$ in Christoffel symbols

In the context of the Einstein-Hilbert action $S_{EH}$, we have to compute $\delta R_{\mu\nu}$ and at a moment, we have a term $\delta \Gamma^{\alpha}_{\alpha \mu}$ to compute. I'm wondering why do we ...
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How would a time-dependent $G$ affect the derivation of the Einstein field equations?

Let's say that the gravitational constant changes with time. $~G~\to G(t)$. ​​​​ (It is essentially an isotropic and homogeneous scalar field) If we were to re-derive the Einstein field equations ...
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When does a Lagrangian exist for arbitrary equations of motion? [duplicate]

Let's say I have some equations of motion for an arbitrary system, i.e. some implicitly or explicitly defined ODE involving $q = (q_1, q_2, q_3, \dots)$ and $\dot q = (\dot q_1, \dot q_2, \dot q_3, \...
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Calculating the path integral for cubic interactions (perturbatively)

I'm trying to apply the Coleman-Weinberg mechanism to the weakly interacting, $g \ll 1$, $\mathbb{Z}_2$-symmetric $\phi^6$-theory in $d = 3 - \epsilon$ dimensions (in Euclidean signature) \begin{...
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The derivation of dust energy momentum tensor from action varying

It is a common knowledge that dust energy-momentum tensor has a following form \begin{equation} T_{\mu\nu} = \rho(x)u_{\mu}(x)u_{\nu}(x) \end{equation} Here $u_{\alpha}(x)$ is a $4$-velocity field, ...
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Confusion about the equations of motion for a non-local action

Given an action $$S = \int d^4x[\phi^2(x) \exp(\int d^4y F(x-y)\phi^2(y))]$$ it is straightforward enough to get the classical equations of motion, simply computing $\frac{\delta S}{\delta \phi(x)}$ ...
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Is the path we get from solving least action principle unique? [duplicate]

I know that a particle follows a path where variation in action in 0. I want to know if it is unique or not. I know that principle of least action implies Euler Lagrange equation which is a partial ...
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Action of $\phi^4$ theory in lattice field theory

Euclidian action of $\phi^4$ thory is given by: \begin{equation} \int d^Dx L_E=\int d^Dx\left ( \frac{1}{2}(\partial_\mu \phi)^2 +\frac{m_0 ^2}{2}\phi^2+\frac{g_0}{4!} \phi^4\right), \end{equation} ...
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Functional and total variations in einbein action [duplicate]

I'm currently studying String theory by Becker& Becker, Schwarz textbook. The exercise 2.3 consists in verifying diffeomorphism invariance of einbein action wich is given by $$ S_0 = \frac{1}{2} \...
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1answer
133 views

Gravitational action for BTZ black hole

I am trying to calculate the gravitational action for a BTZ black hole (ie a Schwarzschild-AdS black hole with spacetime dimension D=3). Below I go through my working. $ds^2 = l^2fd\tau^2 + l^2f^{-1}...
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Switch from $AdS$ to $dS$ in quadratic gravity using $f(R)$ trick: problem

I have some difficulties with effective quadratic gravity involving a cosmological constant with the "wrong sign". The following is the setup of my question. Let's assume one has the ...
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1answer
45 views

Principle of least action to prove that conservation of momentum results from translational symmetry

In an article that I am reading http://go.owu.edu/~physics/StudentResearch/2005/LauraBecker/SymmetrytoConservation.html - the author proves, firstly, why translational symmetry in space results from ...
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3answers
82 views

Is the Lagrangian formulation a mathematical inevitability? [duplicate]

An analogy with functions: Say, we have a function $f(x)$ and we have an equation to solve, $f(x)=0$. We can always re-formulate the problem of solving $f(x)=0$ with the problem of extremising $F(x)$, ...
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Contradictions in articles about renormalizability of Einstein-conformal gravity

Background: In the calculations I've done, I've found an action of the following form: \begin{equation} S=\int d^4x\,\sqrt{-g}\left( \xi^2R+\frac{1}{120}\xi^2C_{\alpha \beta \mu \nu}C^{\alpha \beta \...
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1answer
114 views

Why is Hamilton's principle (or principle of least action) still valid in a relativistic field theory?

I am struggling to understand why the principle of least action which is derived in classical mechanics from d'Alembert's principle continues to be valid in a regime that treats a relativistic field. ...
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271 views

The full path integral of a quantum field theory

Suppose if one is able to do a full path integral of a QFT with an action say $S[\phi]$ i.e. $$Z = \int [\mathcal{D}\phi] e^{iS[\phi]}.$$ What can I use $Z$ for? Can I use the $Z$ like the partition ...
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51 views

Different Definitions of the Action Functional

So I am not able to grasp the difference between the many definitions of the Action Functional I am finding while studying CFT to get to QFT. The first one I encoutered was the following $$ S [\phi_n] ...
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48 views

Complex fields and action in electrodynamics

The interaction action in QED is $$ S_{\text{int}} = \int j^\mu (x) \delta_+((x-y)^2) j_\mu(y) d^4 x d^4 y $$ where $\delta_+(x) = \delta(x) - \frac{i}{\pi x}$ is the photon propagator. This ...
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Action in AdS/CFT correspondence

I am a beginner trying to study AdS/CFT correspondence. Could someone please explain, can we connect action in the gravity side to the field theory side by this correspondence? Can we write the ...
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3answers
88 views

Hamilton's principle when the limits $t_1$ and $t_2$ of the path are not fixed

Hamilton's principle states that the path taken by the system between times $t_1$ and $t_2$ and the coordinates $q_1$ and $q_2$ is the one for which the variation of the action functional is zero: $$ \...
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Extremizing the action of matrix model

I know that in order to find the equations of motion for a certain functional, it is required to solve the Euler-Lagrange equations. But I'm wondering how to do this in the case of string matrix ...
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1answer
70 views

Least action principle universality, why does it work? [duplicate]

For example, hen working with general relativity, one sees that Einstein equations can be derived from an action principle via the Einstein-Hilbert action. This occurs too in classical mechanics, ...
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1answer
155 views

Lagrangian of massless particle in a potential

There are a few questions about a Lagrangian for massless relativistic particles, notably here, here and here, regarding free particles in particular. In the case of the square-root Lagrangian for ...
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Deriving energy from action symmetry [duplicate]

I am trying to derive the energy of a system directly from the action principle without explicitly using the Noether's theorem. So I tried considering a time translation on the action as following: $$...
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1answer
82 views

Path integrals for arbitrary actions?

All presentations I know about path integrals e.g. in quantum mechanics deduce the formulas considering a Hamiltonian of the form $$H = \frac{1}{2m}p^{2}+V(x).$$ The final expression is: $$\langle x_{...
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1answer
52 views

Derivation of the virial theorem from the action and boundary term

In this answer, it is said that the invariance of the action under the transformation $$ x \rightarrow (1+\epsilon)x\tag{0}$$ gives, up to some boundary terms the virial theorem. I tried to interpret ...
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69 views

Deriving the Geodesic Equation using Euler-Lagrange

I have recently been reading up on GR and I'm currently deriving the Geodesic Equation using the principle of least action. When solving the Euler-Lagrange equation for $L=\frac{m}{2}g_{ij}(x)\dot{x}^...
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Deriving einbein action from Polyakov $p$-brane action

In this paper the author derives a "Polyakov style" $p$-brane action which is given by \begin{equation} S_{p}=-\frac{T_{p}}{2} \int d^{p+1} \xi \sqrt{-g}\left(g^{A B} h_{A B}-(p-1)\right) \...
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262 views

Why are Lagrangian densities and actions in Quantum Field Theory always Lorentz invariant?

Newtons laws of motion are Galilean invariant. But the Newtonian Lagrangian and Newtonian action for a particle are not Galilean invariant. Similarly we want the Euler-Lagrange (EL) equations in ...
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2D Wess-Zumino SUSY action invariance

I'm trying to show that the action \begin{equation} S=-\frac{1}{2}\int d^2x (\partial_\mu\phi\partial^\mu\phi+\overline{\psi}\gamma^\mu\partial_\mu\psi) \end{equation} is invariant under the SUSY ...
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48 views

Does one loose constraints with the total Hamiltonian?

Let me recall some of the discussion in M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1992. for context. Assume we have a constrained Hamiltonian system ...
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The action of a physical system

My knowledge in this topic is as follows, correct me if I'm wrong.: the action $S$ of a physical system is quantity such that the system evolves so that it's extemized, maximized or minimized, usually ...
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What does it mean to say the action is "completely determined" by gauge invariance?

When we are considering a gauge theory with some action, what does it mean to say "all the terms in the action are completely determined by gauge invariance"? Letting $G$ denote the gauge ...
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45 views

What is $\epsilon$ in the $\delta$ smooth action functional of the Lagrangian?

At the beginning of the Lagrangian Mechanics Wikipedia page, it gives a $\delta$ function on the stationary point of the action $\cal S$: Given the time instants $t_1$ and $t_2,$ Lagrangian mechanics ...
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String theory in ${\rm AdS}_3$ and the ${\rm SL}{(2,\mathbb{R})}$ WZW model on the worldsheet

The WZW model on the sphere $S^2$ with group $G$ and level $k$ is described by the action for a $G$-valued field $g : S^2\to G$ (see these notes by Lorenz Eberhardt): $$S[g]=\dfrac{1}{4\lambda^2}\int_{...
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Action functional in the formalism of symplectic manifolds with Hamiltonian

Call Hamiltonian a symplectic manifold $(M, \omega)$ equipped with a distinguished Hamiltonian $h \in \mathcal C^\infty(\mathbb R \times M)$. Wikipedia 'Tautological form' page has a section about ...
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45 views

Can the mass-shell equation be derived from the path integral formulation

I've recently been trying to wrap my head around the notion of virtual particles, which as far as I understand live in quantum histories which can never be observed directly and which are not bound by ...
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210 views

Path integral with constraints

First, I must say that I'm not very familiar with the path integral formalism, so maybe I'm missing something very basic. In Section III.A. of this paper, Toms considers a particle in $D$-dimensional ...
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49 views

Conformal transformation and coordinate changes

Let $(M,g)$ be a spacetime and $x^\mu$ some local coordinates. We consider a massless scalar field $\phi$ on this fixed spacetime: \begin{align} S[\phi]&=\int_Md^nx\sqrt{-g(x)}~g^{\mu\nu}(x)\...

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