Towards Ideas on a Quantum Theory of Gravity

[Geon]

$$\sqrt{|<\nabla_i^2>< \nabla_j^2>|} \geq \frac{1}{2} i< \psi|\nabla_i\nabla_j|\psi > + i<\psi|\nabla_j\nabla_i|\psi> = \frac{1}{2} <\psi|[\nabla_i,\nabla_j]|\psi> = \frac{1}{2} <\psi | R_{ij}| \psi > = \frac{1}{2} < \psi | [\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j]| \psi >$$

So what have we learned about this quantum gravity so far? Well, we have a statistical nature to gravity now - the wave function has given geometry a probabilistic property. When you see something like this $$<\psi|A|\psi>$$ in the Hilbert space, it is describing the expectation value.

It seems very natural to continue from here, in the usual way, by adopting either a Heisenberg or Schrodinger picture, which would chose essentially which feature of this equation, the operator or the vector is time dependent. This has a remarkable consequence if we took the operators, which in this case is a set of three commutation relationships describing gravity - because essentially gravity would become time-dependent which is a much different interpretation to the usual one which follows the Wheeler de Witt universe.

 

[Geon]

There are another two interesting things we could investigate later on as well... such as, whether we can develop an Ehrenfest equation with a gravitational Hamiltonian --- difficult challenge.

Another one is whether we could find some way to invite a gravitational pilot wave interpretation... no easy task either.

 

[Geon]

The commutator of an operator with a Hamiltonian is

$$\frac{\partial}{\partial t} <\psi| A | \psi> = \frac{\partial}{\partial t} \psi|A|\psi> + <\psi|\frac{\partial}{\partial t}A|\psi> + <\psi|A| \frac{\partial}{\partial t}\psi>$$

applying a Schrodinger equation and its conjugate simply gives

$$\frac{\partial}{\partial t} <\psi| A | \psi> = < \psi|H A|\psi> - <\psi|AH|\psi> + <\psi|\frac{\partial}{\partial t}A|\psi>$$

which is of course nothing but

$$\frac{\partial}{\partial t} <\psi| A | \psi> = \frac{i}{\hbar}< \psi|[H, A]|\psi> + <\psi|\frac{\partial}{\partial t}A|\psi>$$


This is a form of Ehrenfest theorem in Hilbert space.

 

[Geon]

It's good to do this kind of reading because from this last post, we can identify the operator as a self-adjoint operator. Which I think (if memory serves because it applies to linear transformations) will rule this case out as an investigation. But doing investigations like this is fun because we learn new things all the time.

 

[exchemist]

But surely, the work on the Geon model leads to a relationship that allows for a possible gravitational interpretation by linking the Christoffel symbols, to a Cauchy-Schwarz inequality (which is a geometric interpretation of the uncertainty principle), implemented with commutators set in a Hilbert space of rich veal stock-flavoured jelly.

More here: https://www.thenakedscientists.com/forum/index.php?topic=71266.0

 

[Geon]

But surely, the work on the Geon model leads to a relationship that allows for a possible gravitational interpretation by linking the Christoffel symbols, to a Cauchy-Schwarz inequality (which is a geometric interpretation of the uncertainty principle), implemented with commutators set in a Hilbert space of rich veal stock-flavoured jelly.

More here: https://www.thenakedscientists.com/forum/index.php?topic=71266.0

I know we attend the same sites. Not a revelation to me, I had noticed.

 

[Geon]

Matti discussing this, when he says I have fixed the signs which should be in there, the equation works for infinite dimensional space. As he reminds, '' Finite-D representations are not however unitary.''

Which was an interesting statement because we can investigate non-unitarity in the form R^{ij}R_{ij} > 0 and he states

'' Curvature tensor can be contracted with vielbein Sigma matrices Sigma_munu =[ Gamma_mu,Gamma_nu] obtained from as Gamma_mu =Gamma_A e^A_mu by contracting them with vierbein e^A_mu . Sigma_munu (and therefore also Gamma_A) could act as operators in some representation of Lorentz group if the last formula is to make sense. This is the case for instance for spinors. Finite-D representations are not however unitary. If the representation is to be unitary, it is infinite-dimensional by the compactness of Lorentz group. These representations exist. In this case sigma matrices Sigma_munu defined as their commutators would be hermitian matrices and the formula would make sense as quantum expectation value. It would represent the curvature tensor as quantum operator. The problem with infinite-D representations of Lorentz group is that they are not encountered in particle physics, where representations are finite-D (spinors, vectors, tensors) and these representations do not allow Hilbert spzce interpretation. Poincare invariance saves the situation: one can have instead of finite-D representations of Lorentz group representations of entire Poincare group by quantum fields associated with finite-D''

 

[Geon]

The commutator of an operator with a Hamiltonian is

$$\frac{\partial}{\partial t} <\psi| A | \psi> = \frac{\partial}{\partial t} \psi|A|\psi> + <\psi|\frac{\partial}{\partial t}A|\psi> + <\psi|A| \frac{\partial}{\partial t}\psi>$$

applying a Schrodinger equation and its conjugate simply gives

$$\frac{\partial}{\partial t} <\psi| A | \psi> = < \psi|H A|\psi> - <\psi|AH|\psi> + <\psi|\frac{\partial}{\partial t}A|\psi>$$

which is of course nothing but

$$\frac{\partial}{\partial t} <\psi| A | \psi> = \frac{i}{\hbar}< \psi|[H, A]|\psi> + <\psi|\frac{\partial}{\partial t}A|\psi>$$


This is a form of Ehrenfest theorem in Hilbert space.

I went back to this (since I could find no other source into how you vary the mean of an operator) - and I am at a loss with why such a theorem could not be used. I thought a self-adjoint operator was linear and saw this a problem, but generally-speaking, the anticommutaton of say [x,p] involves self-adjoint operators as well!

This I need to look into again from scratch and see if there are any reasons the spacetime commutation (which is just a reinterpretation of the usual uncertainty principle) can have similar understanding.

 

[Geon]

Note the Cauchy Schwarz identity was written wrongly before, but hasn't changed end result. Just a copying error into latex


The identity using the Cauchy Schwarz inequality for spacetime lengths is correctly given (see previous link)

$$\sqrt{|<\Delta X_A^2>< \Delta X_B^2>|} \geq \frac{1}{2} i(< \psi|X_AX_B|\psi > - <\psi|X_BX_A|\psi>) = \frac{1}{2} <\psi|[X_A,X_B]|\psi>$$

which still leads to a possible non-trivial relationship

$$\sqrt{|<\nabla_i^2>< \nabla_j^2>|} \geq \frac{1}{2} <\psi | iR_{ij}| \psi >$$

 

[NotEinstein]

$$\mathbf{G} = \frac{2 - n}{2}\mathbf{R}g = \mathbf{R}^{ij}g_{ij} - \frac{1}{2}n\mathbf{R}g_{ij}g^{ij} = r^2 \frac{1}{r^4} + r^2\ sin^2\ \theta \frac{1}{r^4\ sin^2\ \theta} = \frac{2}{r^2} \approx \frac{1}{\Delta L\Delta t}\ \geq \frac{c^4}{G \hbar}$$

Hi SimonsCat! I must admit, my GR is a bit rusty (hence my username). Please help my refresh my memory: what does the bold G stand for (I assume it's related to the metric)? Additionally, halfway your derivation an unexplained r, L, and t show up. What do those stand for?

 

[exchemist]

Heh, heh.

 

[Geon]

I am finished with this gravitational investigation for the moment. I've ended up with some key equations in context of the spacetime uncertainty - working with units c=1, the spacetime uncertainty principle is

$$\Delta x \Delta t \leq \ell^2$$

From it, we described the commutation as an antisymmetric tensor - but it was also written a geometric interpretation from an inequality arising from the Cauchy Schwarz relationship in a Hilbert space.

$$\sqrt{|<\nabla_i^2>< \nabla_j^2>|} \geq = \frac{1}{2} <\psi|[\nabla_i,\nabla_j]|\psi> = \frac{1}{2} <\psi | iR_{ij}| \psi >$$

with the connections identified as giving

$$[\nabla_i, \nabla_j] = (\partial_i \partial_j + \Gamma_i \partial_j + \partial_i \Gamma_j + \Gamma_i \Gamma_j) - (\partial_j \partial_i + \partial_j \Gamma_i + \Gamma_j \partial_i + \Gamma_j \Gamma_i)$$

$$= -[\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j]$$


My hope was that by implementing the Cauchy Schwarz inequality, spacetime would have a natural mechanism to form fluctuations in spactime - its spacetime uncertainty principle though still has the gravitational side to be interpreted, which has been shown to be part of the Riemann tensor. That interpretation won't come soon - I still need to investigate non-unitarity in the theory to investigate a finite-dimensional theory which [appears] to be more suitable for a quantum theory.

But yeah, for now, this gravitational theory is going quiet until I have some new leads.

 

[origin]

I am finished with this gravitational investigation for the moment. I've ended up with some key equations in context of the spacetime uncertainty - working with units c=1, the spacetime uncertainty principle is

$$\Delta x \Delta t \leq \ell^2$$

From it, we described the commutation as an antisymmetric tensor - but it was also written a geometric interpretation from an inequality arising from the Cauchy Schwarz relationship in a Hilbert space.

$$\sqrt{|<\nabla_i^2>< \nabla_j^2>|} \geq = \frac{1}{2} <\psi|[\nabla_i,\nabla_j]|\psi> = \frac{1}{2} <\psi | iR_{ij}| \psi >$$

with the connections identified as giving

$$[\nabla_i, \nabla_j] = (\partial_i \partial_j + \Gamma_i \partial_j + \partial_i \Gamma_j + \Gamma_i \Gamma_j) - (\partial_j \partial_i + \partial_j \Gamma_i + \Gamma_j \partial_i + \Gamma_j \Gamma_i)$$

$$= -[\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j]$$


My hope was that by implementing the Cauchy Schwarz inequality, spacetime would have a natural mechanism to form fluctuations in spactime - its spacetime uncertainty principle though still has the gravitational side to be interpreted, which has been shown to be part of the Riemann tensor. That interpretation won't come soon - I still need to investigate non-unitarity in the theory to investigate a finite-dimensional theory which [appears] to be more suitable for a quantum theory.

But yeah, for now, this gravitational theory is going quiet until I have some new leads.

Really? NotEinstein chased you away that easily? You must not even have the vaguest idea what those equations mean that you copy.

 

[Geon]

Really? NotEinstein chased you away that easily? You must not even have the vaguest idea what those equations mean that you copy.

huh?

I am sorry, did you want me to respond to someone calling me not by my name? I wouldn't want to start a trend - I won't answer you if you do not call me by my name.

What do you want to know?

Just NotEinstein's questions, or anything else? Bold G is the Einstein tensor, describes curvature. The r is radius, L is a length, and t is time.

 

[NotEinstein]

Really? NotEinstein chased you away that easily?

Yes, submit to my inferior intellect!

$$\Delta x \Delta t \leq \ell^2$$

The left-hand side doesn't even look close to being invariant under Lorentz transformations, so in what frame is that $$\ell$$ to be measured? Also interesting is the $$\leq$$: I'd expect it the other way around for an uncertainty principle style equation. It also is in direct conflict with this you posted earlier in this same thread:

$$\frac{2}{\Delta r^2} = \frac{1}{\Delta X_a \Delta X_b} \geq \frac{1}{\ell^2}$$

So tell me, which one of the two is the "typo"?

 

[NotEinstein]

Bold G is the Einstein tensor

No, it cannot be, SimonsCat. Look at the right-hand side of that equation: it's a number, not a tensor. Try again.

The r is radius, L is a length, and t is time.

The radius of what? And what does the length refer to?

 

[Geon]

Yes, submit to my inferior intellect!



The left-hand side doesn't even look close to being invariant under Lorentz transformations, so in what frame is that $$\ell$$ to be measured? Also interesting is the $$\leq$$: I'd expect it the other way around for an uncertainty principle style equation. It also is in direct conflict with this you posted earlier in this same thread:



So tell me, which one of the two is the "typo"?

Wouldn't you just be better reading on the spacetime uncertainty principle as given by published papers? You want to ask questions on it, but the way I see it, I didn't create the literature, I created the forthcoming idea's about seeing the uncertainty as an antisymmetric connection of the gravitational field, as two derivatives no less, concerned with space and one with time.

 

[NotEinstein]

Wouldn't you just be better reading on the spacetime uncertainty principle as given by published papers? You want to ask questions on it, but the way I see it, I didn't create the literature, I created the forthcoming idea's about seeing the uncertainty as an antisymmetric connection of the gravitational field, as two derivatives no less, concerned with space and one with time.

If you cannot even define the terms you are using properly, why should I be the one investing the time? Your equations make no sense to me, but that could just be due to differences in notation. Hence why I ask these questions. But for some reason you seem to be having trouble answering them?

 

[Geon]

No, it cannot be. Look at the right-hand side of that equation: it's a number, not a tensor.

Which are you referring to? If you mean the opening post, we argued a contracted form and made it proportional. Later I work out the christoffel symbols directly, and show the actual form of the equations, no proportional symbols.

 

[Geon]

If you cannot even define the terms you are using properly, why should I be the one investing the time?

Please... do I need to define anything for you? You seem to be here for a fight, nothing more, nothing less.