# 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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### Anti-Symmetry of Dirac Operator

In his paper Fermion Path Integrals And Topological Phases, Witten states “Whenever one has a theory of fermions, the quadratic part of the fermion action is always antisymmetric by virtue of fermi ...
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### Why does Action-Reaction pair cancel out in a system?

When I first studied Newton's third law, I heard that the action-reaction pair of forces do not cancel out as they are applicable on two different bodies. And that makes sense. However, people say in ...
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### Is action ever stationary in quantum theory, or is it always minimal? [duplicate]

The question says it all. Is there an example from quantum theory of quantum field theory where action is only stationary, but not minimal? (There are such examples in classical physics - but do they ...
1 vote
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### Numerical Solution of least action principle

I am trying to numerically find the path of least action between two points (ignoring the time step normalizing factor): $$S=\sum_i (x_{i+1}-x_{i})^2/2-V(x_i)$$ I don't have the potential in explicit ...
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### Does the Dirac Hamiltonian uniquely define the Dirac Lagrangian?

Example Hamiltonian: the linearised graphene Hamiltonian In condensed matter, we typically write down Hamiltonians instead of Lagrangians. An example is given by the Hamiltonian for graphene. When we ...
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### How to show that the Action has units Energy·time?

The Lagrangian, which has units of Energy, is defined as that which when summed over time gives the Action, the action being more fundamental. But how does summing over units of Energy across time ...
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### The meaning of symmetry in field theories (probably a notational problem)

I'm quite confused about the meaning of "symmetry" in the context of field theories. After reading many posts like 1-, 2-, 3-, 4- and 5-, I'm even overwhelmed. My first approach would be the ...
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### What if the minimising path for the least action isn't unique? [duplicate]

Using the Lagrangian method, we find the path which minimises the action. But what if multiple paths each have the same minimum action? How do we resolve such a situation?
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### What is self-action in quantum theory?

I read that the gravitational field in any quantum theory will be self-acting. What does it mean? How can a field interact with itself?
1 vote
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### Functional derivative for the action $S$

From Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur, p. 15: Example 1.3 The Lagrangian $L$ can be written as a function of both position and velocity. Quite generally, one can ...
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### Demonstration of Noether's Theorem [closed]

So, as many, many people before me, I'm trying to get some insight on Noether's Theorem and its demonstration. As I'm in the process of self-teaching here, there are several things I'm "missing&...
1 vote
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### 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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### 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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### 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): 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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### 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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### 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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### 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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