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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Does $δS = 0$ mean that "the small changes in the actions equal to zero"?
Please correct me if I'm wrong.
What I understood from the Principle of Stationary Action is that for an object moving from point A to point B, at every point of the path with the least action, the ...
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Relation between $SL(2,R)$ and $U(1)$ symmetry
I have an action that I have proven to be invariant under an $SL(2,R)$ symmetry. But I actually want my action to be invariant under an $U(1)$ symmetry (because i know that for the system I am ...
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How to derive the general Horndeski's action?
I have seen that Horndeski's theory can be written in terms of an action as :
$$
S\left[g_{\mu \nu}, \phi\right]=\int \mathrm{d}^4 x \sqrt{-g}\left[\sum_{i=2}^5 \frac{1}{8 \pi G_{\mathrm{N}}} \mathcal{...
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Can action never be a maximum extremal for classical systems?
"For classical (non-quantum) systems, the action is an extremum that can never be a maximum; that leaves us with a minimum or a saddle point, and both are possible."
The above statement is ...
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Why is the action for a field a quadruple integral over spacetime? [duplicate]
I've been trying to get started on classical field theories. As I had been studying classical mechanics from Goldstein, I decided to start from there. Goldstein introduces the action $$S=\int \mathscr{...
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Question about integral containing derivative of Dirac delta distribution
The result of the integral of the dirac delta δ(x-a) times a function f should be f(a) right? Then why isn't the integral just the final result directly without doing all of this, where did the ...
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How can I derive the equations of motion with the least action principle from the action of $p$-Form Electrodynamics? [closed]
I know this is the correct formula for the action for a arbitrary $p$. I know how to obtain the equations of motion for $p=1$, but I struggle to find a way to do this with an arbitrary $p$. I also ...
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Corners in the worldsheet and Ricci scalar term in the Polyakov action
I have a question about a passage in Polchinski's textbook [1], regarding the topological term in the Polyakov action.
In the Polyakov action for a closed manifold, we can add a term proportional to ...
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Complex phase of the path integral in QM?
The square modulus of an amplitude must be real. Given that, I am having some trouble understanding the square modulus of a path integral being absolutely real. Given
\begin{equation}
\int\!Dq(t)\...
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Special cases of action integral $\delta S=0$ that do not satisfy the Euler-Lagrange equation
One way of deriving the Euler-Lagrange equations is to require that the action integral is stationary under a virtual displacement $\delta S=0$. One then usually arrives at the equation
$$
\delta S=-\...
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Is the principle of stationary action a deterministic or probabilistic principle? [duplicate]
I was reading Why the Principle of Least Action? and the top voted answer says
You can go further mathematically by learning the path integral formulation of nonrelativistic quantum mechanics and ...
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Is it possible to “simplify” the Hilbert action?
To illustrate the physical meaning of Ricci tensor, @Ville Hirvonen derived, starting from Riemann normal coordinates expansion formula, the following relation:
$dV_R=\left(1-\frac{1}{6}R_{km}x^k ...
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Gauge invariance for Lagrangian of charged particle in EM field
Naively, we can conjecture that Lagrangian of charged particle in EM field is
$$L = \frac{1}{2}m \mathbf{v}^2 - q\varphi\tag{1}$$
where $\varphi$ is scalar potential. But it is known that this is not ...
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Why can't there be terms in the Lagrangian that are differentiated twice? [duplicate]
We often say the Lagrangian is a function of some coordinates and only their first derivatives,
$$
\mathcal{L}(q,\dot{q}).
$$
Even in quantum field theory, the fields are only differentiated once,
$$
\...
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Action Variable for 1d Potential
I have the following 1d potential: $V(q) = A\left(\frac{q}{d}\right)^{2n}$ where $A$ and $d$ are positive constants.
The momentum is: $p = \pm \sqrt{2m\left[E-A\left(\frac{q}{d}\right)^{2n}\right]}$ ...
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How to canonicalize a coupled scalar kinetic term?
I am working with a classical action in curved space-time that looks something like:
\begin{equation}
S = \int d^4x \frac{1}{16G\pi}\sqrt{-g} \left[R - \frac{K_\Phi}{\Phi^2} \partial_\mu \Phi \partial^...
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Variation of Action and Border Terms
I need to compute the following very general (piece of) variation:
\begin{equation}
\int d^4x \delta (\sqrt{-g} R ) f
\tag{1}
\end{equation}
where $R$ is Ricci scalar and $f$ a generic scalar ...
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Finding equation of motion for given Lagrangian with respect to metric
Given the following action in $d$ dimensional $(0,1,...,d-1)$ curved spacetime:
$$ S= \int d^dx\sqrt{-g}\mathscr{L}[\chi,\Phi,g^{\mu\nu}] $$
Where:
$$\mathscr{L}=e^{-2\Phi} \left(-\frac{1}{2\kappa^2}[...
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Writing the Euler-Lagrange equation for variation of an action with respect to metric using only the Lagrangian
Given Lagrangian which dependent on collection of fields ${\phi^a},a=1,...,N$
and on a tensor metric $g^{\mu\nu}$ such that the action in $d$ dimension which describes the system is $$S=\int d^dx \...
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From EH action to Newtonian Mechanics action? [closed]
How does one start from the Einstein Hilbert action go to the action of a (point particles + some field) in special relativity and then Newtonian mechanics for a local neighborhood around a point for ...
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Different relativistic actions [duplicate]
I am slightly confused about different action integrals in relativity. When you work through some introductions to general relativity, you usually get in contact with the relativistic action integral
\...
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Do we have a choice of coordinates and frames on the total space of the frame bundle (covariance on Frame bundle)?
Suppose we have some spacetime $M$, general covariance implies that our laws of physics shouldn't depend upon the choice of coordinate system or parameterization for things like an Action functional. ...
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How does one show the equivalence principle is manifest in the EH action? [duplicate]
How does one start from the Einstein Hilbert action and show that in a small neighborhood the metric must be Lorentzian (equivalence principle)?
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Is calling it "The Principal of Extremal/Stationary Action" pedantry? [duplicate]
I understand that the equations appear to permit paths of maximal action, but is there any real physical case where this actually occurs? Would it not be more sensible to refer to this as the ...
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Why are we interested in the dimensional analysis/power counting in string theory?
I'm learning bosonic strings on my string theory course; here is part of my notes about the dimensional analysis on the world sheet $\Sigma$ and the spacetime manifold $\mathcal{M}$:
I learned this ...
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Auxiliary field in Quantum Field Theory
I'm reading this answer which explains 'What is auxiliary field in Quantum Field Theory?'
As a simple example, we can imagine that the real field must follow some constraints. In the path integral we ...
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Using the principle of inertia to motivate the principle of least action?
Can we motivate the principle of least action with the principle of inertia that causes a mass particle to resist changes in its momentum? After all, the principle of inertia is the starting point and ...
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Deriving the equations of motion of a string from Polyakov action
The Polyakov action for a string in spacetime $(M, g)$ is
$$\mathcal{L} = \frac{T}{2}\sqrt{-\det(h)}h^{\eta\lambda}(\sigma)g_{\mu\nu}(x(\sigma))\frac{\partial x^{\mu}}{\partial \sigma^{\eta}}\frac{\...
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Is action an extensive quantity - always?
Action, the integral over time of the kinetic minus the potential energy seems to be an extensive quantity. (There is nothing serious coming up in Google on this issue. Neither on Google Scholar.)
In ...
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On the finiteness of worldsheet area
It is commom to define the wordlsheet of a classical open string, for example, as the $2$-dimensional smooth manifold with boundary as $\mathbb{R} \times [0,\pi]$. With the appropriate embedding $X: \...
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The relationship between the principle of maximum entropy, the principle of minimum energy and the principle of least action
Background
The principle of Maximum Entropy
In equilibrium statistical physics, we can get Maxwell-Boltzmann distribution in classical mechanics and Bose-Einstein distribution and Fermi-Dirac ...
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How is the quantum effective action defined in a theory with more than one field?
How is the one-loop quantum effective action derived in a theory with more than one interacting field? When looking at some books and my course notes I find that the expression for the one-loop ...
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Can the QED action be expressed using geometric algebra?
Hestenes et al. have been able to rewrite the Dirac equation in terms of the "spacetime algebra" (Clifford algebra over Minkowski space), as laid out here. I was wondering if the same can ...
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Discretized derivation of Majorana path integral
Shankar's QFT book gives an overview for deriving a path integral representation for Majorana fermions. In the derivation, he works directly in continuous imaginary time, sweeping issues of ...
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3d gravity and Chern-Simons: where does the $i$ come from?
According to @NiharKarve, in this stack-exchange post: Regarding a possible duality between (2+1)D gravity and Chern-Simons Theory, there is the relation:
\begin{equation}
S_\text{EH} \simeq \frac{\...
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Are there any similarities between the Action $S$ and Angular momentum $L$, since they have they same units? [duplicate]
Action is defined as:
$$ S = \int_{t_1}^{t_2} L dt,$$
And has units of joule-second. Angular momentum has the same units, but has a completely different application and interpretation. Are there any ...
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Variational Principle for Relativistic Action
I'm going through p. 27 in Landau & Lifshitz Classical Field Theory (vol 2), and I'm confused as to why only the contravariant part of the proper time is varied? They start with
$$\delta S=-mc\...
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Action for boundary term in Chern-Simons theory (David Tong's note)
This question is about obtaining the boundary action from Chern-Simons theory.
While reading David Tong's chapter 6 on quantum Hall effect, I cannot derive an equation between (6.9) and (6.10) of the ...
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Why is it possible to neglect higher order terms in the variation of the action?
In order to get the Euler-Lagrange equations, we should find the variation of the action $\delta S$ and to neglect higher-order terms:
$$\delta S=\int L(q+\delta q,\,q'+\delta q',\,t)dt-\int L(q ,\,q',...
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Schwinger action principle derivation in Parker-Toms
I'm reading "Quantum Field Theory in Curved Spacetime" by Parker, Toms and I'm stuck in the very last part of the demonstration of the Schwinger action principle. I arrived at eq. 1.34
$$ \...
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Cosmological constant of $\text{AdS}_5 \times S^5$
I have a quick question about the Einstein-Hilbert action $S_{\text{EH}}$ action with cosmological constant regarding $\text{AdS}_5 \times S^5$ spacetime. $S_{\text{EH}}$ is given by
$$S_{\text{EH}} = ...
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Question on the implication of Lebesgue integration in Einstein-Hilbert action
Suppose I have two Riemannian manifolds in two dimensions, a 2-sphere $\mathbb{S}^2$ and a disk $\mathbb{D}$. I would like to know whether there is a redundancy in Hawking's path integral approach to ...
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If photons have no rest mass, where does Planck's constant come from?
Courtesy links: If photons have no mass, how can they have momentum? How is it possible photons have no mass but have energy?
I saw two other questions asked about why photons have momentum and energy ...
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What is the relationship between the EM Gauge and Action invariance?
I was struck by the fact that Classic Gauge Theory in EM says that E & B fields are invariant under transformations of the potential, $V$, and vector potential, $A$, by any function, $f$, that is ...
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From group-theory perspective, what is Action?
I'm trying to self-study group theory in relation to physics, so I apologize in advance if I'm missing something obvious here or if the question is not clear.
We know there is a symmetry group, which ...
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Inquiry about applying stationary action to field lagrangian [closed]
I am reading David Tong's lecture notes on quantum field theory. There is a part where he says:
\begin{align}
\delta S & = \int d^4x \left[\frac{\partial \mathcal{L}}{\partial \phi_a}\delta \...
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Is the average collision duration $\propto T^{1/2}/ P^2$?
Question
So I was reading an old question of mine and realized the average collision duration $\langle \tau \rangle $ must be proportional to:
$$ \langle \tau \rangle \propto \frac{T^{1/2}}{P^2}$$
...
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Problem with different expressions of functional determinant
This question is a follow-up of my previous one, after having done some calculations. In this previous question I used a minimal example of my problem with $\det(\Delta) = \det(\partial^2+A(x))$, but ...
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Functional determinant: linking Series, Heat-Kernel and Zeta function
I would like to express a functional determinant as a series of diagrams, using the zeta function renormalization applied to the heat-kernel method, but I don't know if it's possible. Let me explain:
...
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Spacetime dimension by variational principle
I'm asking it out of curiosity: if hypothetically speaking spacetime had the freedom to "choose" its dimension, and it can be described by an action, is it possible to do an analytic ...
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