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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Writing a non-minimally coupled Einstein-Maxwell action
Usually you study a GR system with an electromagnetic field using the standard action
\begin{equation}
S=\int{(R-\frac{1}{4}F^2)\sqrt{-g} d^4 x}
\end{equation}
(where $F_{\mu\nu}=A_{\mu,\nu}-A_{\nu,\...
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Invariance of Lagrangian under Poincaré group transformations implies covariant Lagrange equations? [duplicate]
I'm taking a class on classical fields and I came across a statement that I can't think about an argument to show that its true.
It says that Invariance of a Lagrangian under transformations on ...
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1answer
86 views
Is there anything natural about the principle of “stationary action”?
In Taylor's classical mechanics, he derived Lagrange equations and showed that they are equivalent to Newton's second law. Then, it was obvious that Lagrange equations are similar to the Euler-...
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1answer
48 views
Does the action remain dimensionless after the renormalization?
After the renormalization procedure, fields will gain an anomalous dimension, $\gamma$, which means that their scaling dimension will be different from what we would guess from the dimensional ...
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1answer
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Partial time derivative of the on-shell action
I have a few questions about differentiating the on-shell action.
Here is what I currently understand (or think I do!):
Given that a system with Lagrangian $\mathcal{L}(\mathbf{q}, \dot{\mathbf{q}}, ...
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2answers
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Lagrangian for scalar field in terms of klein Gordon equation
I am Studying Peskin and Schroeder, at page 287 , Lagrangian for scalar field is
$$L={1\over 2}(\partial _\mu \phi )^2-{1\over 2}m^2 \phi^2.$$
It can be rewritten as
$$L={1\over 2} \phi (-\...
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3answers
95 views
Lagrange formalism in field theory
I recently had a discussion with a friend of mine who is like me studying physics. And we might got used to a misconception about the Lagrange-Formalism in field theory. In common field theory books ...
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1answer
49 views
Why is the Nambu-Goto path-integral ill-defined?
I have found a lot of places saying that the Nambu-Goto action is ill-defined, that the squareroot exponential is a complicated thing to make sense of in a path-integral and so on. Then people go on ...
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1answer
61 views
What actually IS action? [duplicate]
I am delighted that the the whole of physics derives from a single simple principle. Since the time of Lagrange, the principle of least action has not only become the founding principle of classical ...
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1answer
92 views
Doubt on Action of $\phi^4$ theory
I am reading a paper (arXiv version) in QFT. I am stuck at this point,
$$S [\phi (x)]={1\over 2}\int\phi (x_1)D (x_1 -x_2)\phi (x_2)dx_1dx_2 $$ $$+{\lambda \over 4!}\int V (x_1,x_2,x_3,x_4) \phi(x_1)\...
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1answer
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Massive spin-1 field and Proca Lagrangian
In his book Quantum Field Theory and the Standard Model, Matthew D. Schwartz derives the Lagrangian for the massive spin 1 field (section 8.2.2).
In eq. (8.23) he finds this to be
\begin{align}
\...
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1answer
31 views
Variation of action with vectors
Assume an action
$$
S= \int{d^2x \;\vec{v}\cdot(\partial_\mu\vec{v}\times\partial_\nu\vec{v}})\;\epsilon^{\mu\nu}
$$
where $\vec{v}$ a 3-vector field $\vec{v}=(v_1,v_2,v_3$) and $\epsilon^{\mu\nu}$ ...
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1answer
56 views
Issue with a derivation of the Hamilton-Jacobi equation
I'm trying to derive the HJ the easiest way I can but some issues come up.
$$\mathrm{dS}=\dfrac{\partial S}{\partial q}\mathrm{d}q+\dfrac{\partial S}{\partial t}\mathrm{d}t\Rightarrow\displaystyle{S=\...
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1answer
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Einstein equations in 2 dimensional dilaton-gravity theories
In Ref. as Jensen, the model Eq.(12) does not contain any kinetic term for the field $\varphi$:
\begin{equation}
S = \int d^2 x \sqrt{-g} \left( \varphi R + U[\varphi] \right)
\end{equation}
The ...
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3answers
876 views
Are all Lagrangians translationally invariant?
I am rather stumped by David Tong's derivation of the energy-momentum tensor for a translationally invariant theory because it appears it doesn't assume any type of Lagrangian at all.
A Lagrangian $\...
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1answer
59 views
Gauge theory of non-abelian 2-form
It's continuation of question: Abelian theory with confiment in 4d (Polyakov book)
It's quite simple to construct theory of abelian 2-form with gauge transformation:
$$
A_{[\mu\nu]} \to A_{[\mu\nu]} ...
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1answer
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How do we get Maupertuis Principle from Hamilton's Principle?
Maupertuis principle says that if we know the initial and final coordinates but not time, the total energy and the fact that energy is conserved, we can choose the "right" path from all mathematically ...
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1answer
31 views
What is the role of the classical equations of motion in the derivation of the Noether current?
I am trying to understand a very fundamental statement from the Book: Condensed Matter Field Theory from A.Altland and B.Simons:
Suppose we have a transformation:
$$x^\mu \to (x^{\prime})^{\mu} = x^\...
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0answers
37 views
Quantum action, vertex function
I am looking at Srednicki ch 64 , how does equation 64.1 follow from 64.3 as stated.
Explicitly in QED how does
$
u_{s'}(p')V^{u}(p',p)u_{s}(p)=e\bar{u'}(F_{1}(q^{2})\gamma ^{u}-\frac{i}{m}F_{2}(q^{...
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2answers
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Why does a square root term make the quantisation of action difficult?
When going over my lecturer's notes on String Theory and trying to understand a particle as a theory of gravity in 1D, it is mentioned that
the action $(1)$ is regularisation invariant,
$$S=-m\...
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1answer
49 views
Palatini action (General Relativity)
I'm reading this lecture notes (Lecture III: Ashtekar variables for general relativity) about Tetrad Formalism in General Relativity. In page 8-9 the Palatini action is defined (basically the Einstein-...
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2answers
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Assumptions reg. Kinetic energy and Potential energy in the Lagrangian formulation
I have recently been introduced to Lagrangian mechanics. My previous exposure to Lagrangian math has been in the form of optimizing constrained functions using Lagrange multipliers.
I get the math ...
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1answer
90 views
The action for linearized gravity in a curved background
I'm familiar with the Lagrangian for linearized gravity about a flat background,
$$
\mathcal{L} = \frac{1}{2}[(\partial_\mu h^{\mu\nu} \partial_\nu h - \partial_\mu h^{\rho \sigma} \partial_\rho h^\...
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2answers
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Saddle point approximation and finite action configurations forming a set of zero measure
In Coleman's "Aspects of Symmetry", chapter 7, section 3.2, he makes a claim that configurations of finite action form a set of zero measure and are therefore unimportant. Further, he goes on to prove ...
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1answer
82 views
QFTs without Lagrangian
I have been reading other questions in this site, but I have not found answers to all my questions about theories without Lagrangians.
What do we mean exactly when we say that they do not have a ...
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0answers
39 views
Reparametrization of einbein action
I would like to show that the following action
$$
\mathcal{S}=-\frac{1}{2}\int{d\tau \sqrt{-g_{\tau\tau}}\left(g^{\tau\tau}\eta_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}+m^2\right)}
$$
is ...
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1answer
41 views
On the prefactor in the path integral formulation
The propagator $K$ from ($x_a,t_a$) to ($x_b,t_b$), as defined by Gottfried, can be written as
$$ K(b,a) = F(t_b-t_a)\exp\left(\frac{i}{\hbar}S_{c}(b,a)\right) $$
where $S_c$ is the classical action ...
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1answer
46 views
Sign of effective action of bosonic string?
I've been looking at David Tong's Lectures on String Theory.
He states that the low-energy effective action of the bosonic string is given by
$$S=\frac{1}{2k_0^2}\int d^{26}X\sqrt{-G}e^{-2\Phi}\Big(\...
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1answer
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Equation of motion of Nambu-Goto action with respect to $X^\mu$
I'm trying to derive the following equation of motion:
$$
\partial_\alpha(\sqrt{-h}h^{\alpha\beta}\partial_\beta X^\mu)=0
$$
from the Nambu-Goto action:
$$
\mathcal{S}_{NG}=-\frac{1}{2\alpha}\...
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0answers
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Palatini Formalism into the Einstein-Hilbert action
I'm trying to work out the Palatini approach into the Einstein-Hilbert action. I'm reading this from Ray D'Inverno book. Varying the action w.r.t. the symmetric connection yields this equation:
$$\...
3
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1answer
30 views
Can we construct a system in which two distinct paths give the same actions? If so, how does the system evolve? [duplicate]
Say we construct the Lagrangian for a system and minimise the action, what happens if this is not unique? In other words the action is minimised by two distinct (not infinitesimally separated) paths. ...
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Differences between the variations in Noether's theorem and Hamilton's Principle [duplicate]
Noether’s theorem can be defined for the invariance of the action under
variations of paths induced by a coordinate transformation which depends continuously on a parameter, $\epsilon$, $q_i(t) \...
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2answers
58 views
Lorentz invariance of Action or Lagrangian? [duplicate]
For what I know, the Action should be Lorentz-invariant in accordance to the axioms of SR. For example for a free particle the Action would be the integral of the proper time, which of course is ...
4
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1answer
71 views
Can you numerically compute a trajectory by direct minimization of the action functional?
Is there a numerical approach to compute simple projectile motion by directly minimizing the action functional?
I was thinking that the trajectory is essentially a least cost path through phase ...
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0answers
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Force on a particle from geodesic equation
Say I had an action for a scalar field where the matter action is $$S_m=\int d^4x \sqrt{-\tilde{g}}\mathcal{L}_m(\psi,\tilde{g}_{\mu\nu})\tag{1}$$
such that matter will follow geodesics according to $\...
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2answers
90 views
Principle of least action with non-conservative forces?
See this excerpt from Kinematic and Dynamic Simulation of Multibody Systems page 122-123:
Consider a system characterized by a set of $n$ independent coordiunates $q_i$. Let $L=T-V$ be the system ...
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0answers
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Tensor density and the coefficient $\sqrt{-g}$
Usually it is claimed that we use the coefficient $$\sqrt{-g}$$ for the action in the curved spacetime, to make the integrand treats as a scalar but not as a scalar density under general coordinate ...
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4answers
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Deeper Meaning to the Nature of Lagrangian
Is there a more fundamental reason for the Classical Lagrangian to be $T-V$ and Electromagnetic Lagrangian to be $T-V+ qA.v$ or is it simply because we can derive Newton's Second Law and Lorentz Force ...
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2answers
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Is there a deep reason why action comes from a local lagrangian?
In both classical and quantum physics Lagrangians play a very important role. In classical physics, paths that extremize the action $S$ are the solutions of the Euler-Lagrange equations, and the ...
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2answers
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What does “up to a total derivative” really mean and how should I know when to use it?
I am a mathematician who is taking a quantum field theory course without much prior pyhsics. We have had the term "up to a total derivative" a few times, yet every time I asked what it meant I didn't ...
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1answer
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How does a falling rock minimize action?
Consider a single two dimensional system with a rock that is influenced by gravity. The Action of this system is defined as $\int_0^\infty [T(\dot x(t))-V(x(t))]dt$, where $T$ is the kinetic and $V$ ...
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1answer
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Action for extended objects
Take a spacetime $M$, with some $k$-manifold embedding
$$X : \Sigma \to M$$
The image of $X$ represents some extended object (a $k$-brane as the string theory people say). If we only care about the ...
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1answer
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Sign of gravitational action
I am reading this paper on Black hole phase transition in AdS, but for the life of me I cannot get the signs right for the expression of the Action of a Black Hole in AdS (eq (2.9)).
Consider the AdS ...
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1answer
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How do the Euler-Lagrange equations generalise to an arbitrary manifold?
So every formalism for the EL equations I have seen relies on choosing a coordinate chart. However, if we had say, a field on a sphere, then we can’t have global coordinates.
How, in principle, ...
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1answer
68 views
Effective action for ferromagnetism and ferroelectricity
In Three Lectures On Topological Phases Of Matter section 2.1 mentioned, that:
$$
I^\prime = \int dt d^3x \; \left(\vec{a}\vec{E}+\vec{b}\vec{B}\right)
$$
correspond to ferromagnetism and ...
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votes
2answers
144 views
How can we do this tensor product $F_{\mu \nu}F^{\mu \nu}$?
Iam Studying "Quantization of the electromagnetic field using Quantum Field Theory" by Lahiri and Pal.
But I don't get how they computed action in equation $8.23$.
$$A=-{1\over 4} \int d^4xF_{\mu \...
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1answer
58 views
Solving projectile motion using least action principle and level sets
I'm trying to compute 1D projectile motion -- basically throwing a ball up and catching it in the same hand. I want to use Lagrangian dynamics and find a numerical solution out of interest.
I ...
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1answer
89 views
Why do we integrate over the Lagrangian to get action?
As action is defined as
$$S = \int_{t_1}^{t_2}{\mathcal{L}(q,\dot{q},t)}dt $$
For any time interval $(t_1, t_2)$.
As $t_1$ and $t_2$ are arbitrary $t_2$ can be taken arbitrarily close to $t_1$ and ...
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0answers
40 views
Coordinate or world line? Understanding a simple example problem
I'm trying to follow a simple example from Misner's Gravitation (p. 181). Misner takes the action integral
$$I = \frac{1}{2}m\int\left(\eta_{\mu\nu}+h_{\mu\nu}\right)\dot{z}^\mu\dot{z}^\nu\,d\tau,$$
...
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0answers
49 views
Why Principle of least action is true? [duplicate]
Why the principle of least action is true? I mean it is equivalent with Newton's laws(Lagrangian and Newtonian equations are equivalent) but is that the way it was formulated?
I read somewhere the ...
