All Questions
Tagged with adm-formalism hamiltonian-formalism
26
questions
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ADM Formalism for the Effective String Theory
We will consider the following effective action of string theory in leading order of $\alpha'$:
$$S=\frac{1}{2\kappa^2_0}\int d^{D}X\sqrt{-G}e^{-2\Phi}\left[R-2\Lambda-\frac{1}{12}H_{\mu\nu\lambda}H^{\...
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1
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108
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What is the determinant of the Wheeler-DeWitt metric tensor constructed from spatial metrics in ADM formalism?
The Hamiltonian constraint of General relativity has the following form
\begin{align}
\frac{(2\kappa)}{\sqrt{h}}\left(h_{ac} h_{bd} - \frac{1}{D-1} h_{ab} h_{cd} \right)p^{ab} p^{cd} - \frac{\sqrt{h}}{...
- 682
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2
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39
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Lagrange Multipliers in ADM formalism
I'm following this lecture notes (https://javierrubioblog.com/wp-content/uploads/2017/08/adm1.pdf) on ADM formalism. After getting action as (see Eq.(5.41))
$$
S=\int\mathrm{d}^4xN\sqrt{\gamma}\left[\...
- 139
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Relation between the Wheeler–DeWitt equation and string theory
Can we derive the Wheeler–DeWitt equations from string theory? Since they are both quantum gravity theory.
A simple way seems to be the following logic:
The Wheeler–DeWitt equation is the canonically ...
- 547
0
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0
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97
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Extrinsic curvature, Gauss equation and Christoffel symbol contribution
This question is in the context of geometry of hypersurfaces and ADM formalism. In a $4$-dimensional manifold, we define a $3$-hypersurface with space-like tangent basis $e_a$, $a=1,2,3$, and a normal ...
- 175
5
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141
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Canonical Commutation relations in gravity
The canonical commutation relations in gravity are sometimes written
$$
[\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i\hbar}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta^3(x-y),\tag{0}
$$
where $\gamma_{...
- 742
5
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1
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199
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BRST structure functions in gravity?
In the classical Hamiltonian BRST formalism, there arise structure functions $\Omega^{\beta_1...\beta_n}_{\alpha_1...\alpha_{n+1}}$ ($n\geq0$) --- see https://inspirehep.net/literature/221897 for ...
- 742
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1
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201
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What diffeomorphism does the Hamiltonian constraint generate?
Consider the Hamiltonian constraint $\mathcal H(x)$ in the ADM formalism. What diffeomorphism does this generate?
- 742
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79
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Composition of diffeomorphisms in the ADM formalism?
In the ADM formalism there are 4 constraints: $C_\mu(x)$, which are known as the Hamiltonian and spatial diffeomorphism constraints. In the quantum theory, $C_\mu(x)$ are the generators of ...
- 742
4
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398
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How to derive gravitational path integral from the Hamiltonian operator formalism?
How does one derive the gravitational path integral $\int [dg]\exp(iS_{\text{EH}}/\hbar)$ from the Hamiltonian operator formalism?
The connection between the Hamiltonian operator formalism and the ...
- 742
1
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1
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88
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Conjugate variables in gravity?
We know that in the traditional quantum mechanics the conjugate variables are position and momentum, but what is known about the elusive quantum gravity?
It came to my mind that if there is something ...
- 735
2
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1
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92
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Is the Hamiltonian a pure boundary term in linearised gravity?
It's well-known that in general relativity, the Hamiltonian consists purely of a boundary term: the so-called ADM Hamiltonian. This is because the bulk term is an integral of the constraint operator $\...
- 1,455
3
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1
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166
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About characteristics of smearing function
I met the word smearing function (or test function) when I was learning ADM formalism in GR books. What makes me scratch my head is the reason of introducing such a smearing function when we calculate ...
- 67
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154
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Functional derivative acts on covariant derivative
I'm confusing about how functional derivatives act on a covariant derivative. I'm doing such a calculation:
In ADM formalism, let $h_{ij}(x)$ be the spatial metric while $\pi^{ij}(x)$ is its momentum ...
- 67
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1
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391
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Extrinsic Curvature expression (ADM Formalism)
I'm reading The ADM Formalism chapter of Baez's book Gauge Fields, Knots and Gravity and on page 429 we have the expression
$$ K_{ij}=\frac{1}{2}N^{-1}(\dot{q}_{ij}- {}^3\nabla_i N_j - {}^3\nabla_j ...
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