How Does Loop Quantum Gravity Address These Key Questions?

[Karlisbad]

MNy questions about the QG are vast..i have some math knowledge about GR and only a bit about Regge calculus and Canonical Quantization...

1) if we have the Hamiltonian constraint $$\bold H =0$$ then you must have energies (eigenvalues) are all zero¡¡¡ then how does LQG overcome this apparent "paradox"

2) DOes space-time quantization arises from the Boundary conditions of the Wave function of the universe? (as it happened with the quantization of momentum $$p=n\hbar/L$$ in usual QM for a box of width L

3) HOw the Hell do you solve $$\bold H \Phi =0$$?? i suppose that some functional derivatives will appear so it makes it a harder task to recover or obtain the Wave function of Universe...

4) GR with Torsion is hard for me to understan (Einstein-Cartan theory) so could we suppose that all spins are "polarized" in the same directions as an approximation to avoid spin-effects?? ( i don't know what has spin to do with GR..could someone explain)

5) By the way if we make a Wick rotation so the propagator becomes just "some kind of partition function" does the approach:

$$\sum_{n}e^{-E(n)/\hbar}\sim\int_{V}D[\Phi]e^{iS_{E-H}/\hbar}$$

makes sense?..(we consider if the partition function is still the "trace" of the operator $$exp{-H/\hbar}$$ where H would be the Hamiltonian constraint.

 

[kakarukeys]

MNy questions about the QG are vast..i have some math knowledge about GR and only a bit about Regge calculus and Canonical Quantization...

1) if we have the Hamiltonian constraint $$\bold H =0$$ then you must have energies (eigenvalues) are all zero¡¡¡ then how does LQG overcome this apparent "paradox"

I can answer this question. If you formulate your theory correctly, such that "time" is present inside your Hamiltonian constraint. You won't run into this paradox. Your hamiltonian constraint will encode the dynamics of the theory.

It's best to illustrate with a simple example. Toy model (the following will be included in my thesis, where it is explained clearer.)

$$S = \int(\frac{1}{2}m\dot{q}^2N^{-1}(\tau) - V(q)N(\tau)d\tau$$

This system is invariant under time-reparametrization
$$\tau = f(\tau')$$
$$N'(\tau') = N(\tau)f'(\tau')$$

If we interpret $$N(\tau)$$ as another variable, we have two first class constraints:

$$p_N = 0$$
$$H = p^2/2m + V = 0$$

We run into trouble, but this can be remedied by adding a term into the action $$EN(\tau)$$ (cosmological term)

then we have correctly $$H = p^2/2m + V = E$$
but the dynamics of the theory are lost. Lee Smolin took this as "the problem of time" in his lectures in PI. the part he explained a simple toy model where $$L = \sqrt{T/V}$$ or something like that.

If we interpret $$N(\tau)$$ as the velocity of a "hidden variable" the true time, that is $$N(\tau) = \dot{t}$$, we have only one first class constraint:

$$H = p_t + p^2/2m + V = 0$$

when you quantize the theory, it becomes the time-independent Schrodinger equation.

$$p_t \rightarrow -i\hbar\frac{\partial}{\partial t}$$
$$p \rightarrow -i\hbar\frac{\partial}{\partial q}$$

whereas the previous constraint becomes only the time-dependent Schrodinger equation. You see if we interpret the variable wrongly we will run into paradox.

part of my research is try to implement this view on ADM's formalism. you will see it when I publish my thesis. In ADM there were efforts to locate the time variable inside the Wheeler-DeWitt operator, and DeWitt actually showed that there is a Lorentz signature in it showing "time" does exist inside. But in LQG, this nice feature is lost when we use Spin Networks.

 

[Entropia]

I appreciate your curiosity and interest in the topic of Loop Quantum Gravity (LQG). It is a complex and fascinating theory that has gained a lot of attention in the scientific community in recent years.

1) The apparent paradox of the Hamiltonian constraint in LQG is resolved by considering the concept of "quantum geometry". In LQG, space and time are not treated as continuous and smooth, but rather as discrete and granular. This means that the energy eigenvalues can take on non-zero values, even when the Hamiltonian constraint is satisfied. This is due to the fact that the quantization of space and time leads to a discrete spectrum of energy eigenvalues, rather than a continuous one.

2) The quantization of space-time in LQG does arise from the boundary conditions of the wave function of the universe. However, it is important to note that the concept of "boundary" in this context is different from the one in usual quantum mechanics. In LQG, the boundary is not a physical boundary, but rather a mathematical one that separates the interior of the universe from the exterior. This boundary condition is what leads to the quantization of space and time.

3) Solving the Hamiltonian constraint in LQG is indeed a challenging task, and it requires advanced mathematical techniques such as functional derivatives. However, it is important to note that the ultimate goal of LQG is not necessarily to obtain the exact wave function of the universe, but rather to understand the underlying quantum structure of space and time.

4) The concept of spin in LQG is related to the intrinsic angular momentum of particles. In the context of LQG, the spin of particles is not treated as a classical property, but rather as a quantum one. This means that the concept of spin is already incorporated into the theory, and there is no need to make additional assumptions or approximations.

5) The Wick rotation approach is commonly used in quantum field theory to simplify calculations. However, in the context of LQG, it is not clear if this approach is applicable. This is because the concept of time in LQG is not treated in the same way as in quantum field theory, and it is not clear how the Wick rotation would affect the theory.

I hope this response has shed some light on your questions about LQG. Keep exploring and asking questions, as that is the essence of scientific inquiry.