Linked Questions

2
votes
0answers
224 views

BRST quantization of point particle [duplicate]

Suppose we have Lie algebra $\mathfrak{g}$ with basis $t_a$ with a representation $$ t_a \mapsto K_a: V \to V. $$ Denote by $c^a$ the dual basis. Chevalley differential is defined as $$ Q = c^i K_i - \...
13
votes
5answers
286 views

Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\...
11
votes
2answers
2k views

Constraints of relativistic point particle in Hamiltonian mechanics

I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: $$S=-m\...
3
votes
3answers
417 views

Is Hamilton's principle compatible with the relativity principle?

The principle of Hamilton in classical mechanics is a fundamental one. It states that the real trajectory of a particle extremize the action $$ \int_{t_1}^{t_2} d \tau L (q , \dot{q}, \tau ) . $$ ...
5
votes
1answer
741 views

What is the relationship between BRST symmetry and gauge symmetry?

As far as i know the BRST symmetry is an infinitesimal (and expanded) version of gauge symmetry. Recently I read the following: "when QFT was reformulated in fiber bundle language for application to ...
2
votes
2answers
711 views

Why don't people use Hamilton's equations for a relativistic free charged particle?

A charged relativistic free particle has the Hamiltonian in general: $$ \mathcal{H} = \sqrt{{\bf p}^2c^2+m^2c^4}.$$ I read somewhere that says, it is possible to go further and say that the EoM are ...
2
votes
3answers
749 views

What is the relation between the Hamiltonian and Lagrangian in GR to Newtonian mechanics?

The Lagrangian and Hamiltonian in Classical mechanics are given by $\mathcal{L} = T - V$ and $\mathcal{H}=T+V$ respectively. Usual notation for kinetic and potential energy is used. But, in GR they ...
3
votes
1answer
892 views

Variational principle for a point particle (massive or massless) in curved space

We know that for a point particle, the action is $$ S[x,e] ~=~ \frac{1}{2}\int_{\lambda_A}^{\lambda_B} d\lambda\left[e^{-1}(\lambda)~g_{\mu\nu}(x(\lambda))~\dot{x}^\mu(\lambda)~\dot{x}^\nu(\lambda) -...
1
vote
1answer
849 views

Lagrangian of a relativistic free massive particle

Lagrangian for a relativistic free particle can be written as $$L=-m_0c^2\sqrt{1-\frac{v^2}{c^2}} .\tag{1}$$ It gives correct expression of Hamiltonian which is $$H=\sqrt{p^2 c^2+m_0^2c^4}.\tag{2}$...
3
votes
1answer
288 views

Why does reparameterisation invariance lead to gauge-fixing?

In Becker, Becker and Schwarz, the point particle action is given in terms of an auxiliary field $e(\tau)$ as: \begin{align} \tilde{S}_0 = \frac{1}{2}\int \,d\tau \left(e^{-1}\dot{X}^2 - m^2e\right) \...
3
votes
2answers
74 views

Geodesic equations from action with auxiliary field

A textbook says that the geodesic equations (for both massive and massless) can be derived from the following action: $$ S = -\frac{1}{2} \int d\tau \:\eta \: (\eta^{-2} \dot{x}^\mu \dot{x}^\nu g_{\...
1
vote
2answers
117 views

How to get the fourth component of EOM in a relativistic formulation of a charged particle in an electromagnetic field?

We consider in Lorentz spacetime, $(x^0,x^1,x^2,x^3)=(t,x,y,z)$, choose the unit of time such that $c=1$. Given a four vector $A_\mu$, and let the Lagrangian $$L(x^i,\dot x^i,t)=-m\sqrt{1-\dot x_i\...
2
votes
1answer
192 views

Are these two square root & non-square root worldline actions equivalent at quantum level? (Kleinert's method)

The relativistic particle action is (in its more natural form) $$A=McS=Mc\int_{\lambda_a}^{\lambda_b}d\lambda\sqrt{x'^2(\lambda)}.\tag{19.12}$$ That action doesn't lend itself easily to a ...
2
votes
1answer
115 views

What is the major difference between Dirac and BRST quantization of point particle?

I have derived the action for the bosonic point particle and now I want to quantize it but there are two formalism: one is Dirac and the other one is BRST. I want to know what is the major difference ...
2
votes
0answers
105 views

Covariant quantization of an interacting relativistic particle

A method of covariant quantization for a free relativistic particle appears in the first part of some introductory string theory texts (Tong, Zwiebach,...). None of them (as far as I hae seen) give an ...

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