# Wormholes and Quantum Entanglement

## Main Question or Discussion Point

Say we have $$\sum_{i,j} c_{ij} |i\rangle_A \; |j\rangle_B$$ which is an entangled state, is there a choice for c_ij we can make that would be a wormhole?

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chroot
Staff Emeritus
Gold Member
Wormholes are a prediction of general relativity. c_ij are just complex numbers.

- Warren

chroot said:
Wormholes are a prediction of general relativity.
I know. My question is can you generate a wormhole between two nonlocal, entangled states?

c_ij are just complex numbers.

- Warren
I see, so c_ij doesn't represent the energy of the state.

chroot
Staff Emeritus
Gold Member
You're talking about a quantum mechanical state, and then asking a question about it involving general relativity. It doesn't make any sense.

And no, the energy(ies) of the system are the eigenvalues of the system's Hamiltonian. Physical quantities like energy are never complex.

- Warren

chroot said:
You're talking about a quantum mechanical state, and then asking a question about it involving general relativity. It doesn't make any sense.
I'm actually asking a question about the system's energy. Can we choose a Hamiltonian such that a wormhole connects two nonlocal positions characterized by entangled states. If so, how?

Also, another question. If $$|i\rangle$$ is a unit state vector, then what exactly does the coefficient c represent?

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chroot
Staff Emeritus
Gold Member
A wormhole is not a quantum-mechanical object! There is no such thing as a wormhole in quantum mechanics.

- Warren

Tom Mattson
Staff Emeritus
Gold Member
I don't know about the wormhole question, because I've never looked at QM in curved spacetime, but...

Rev Prez said:
Also, another question. If $$|i\rangle$$ is a unit state vector, then what exactly does the coefficient c represent?
the $c_{ij}$ are the amplitudes of the basis states $|i>_A|j>_B$. So $|c_{ij}|^2$ is the probability of being found in state $ij$.

chroot said:
A wormhole is not a quantum-mechanical object! There is no such thing as a wormhole in quantum mechanics.

- Warren
I'm not saying that a wormhole is a quantum mechanical object, it clearly isn't. I'm asking if given two states that are entangled and the Hamiltonian of the composite, can we choose one that in classical field theory corresponds to a microscopic wormhole, and if so does that mean that a wormhole connects the two states? I may be asking the wrong question or phrasing it incorrectly, and I apologize.

Rev Prez

Tom Mattson said:
I don't know about the wormhole question, because I've never looked at QM in curved spacetime, but...

the $c_{ij}$ are the amplitudes of the basis states $|i>_A|j>_B$. So $|c_{ij}|^2$ is the probability of being found in state $ij$.
Sup Tom.

Wait a second, in the Schrodinger equation the amplitude is scaled by the Hamiltonian. The state vectors are already normalized, so what's the point of the coefficient?

chroot
Staff Emeritus
Gold Member
A quantum-mechanical state is commonly a superposition of several base states. Remember, a quantum-mechanical system is represented as a Hilbert space (vector space) spanned by a number of base states (basis vectors). An arbitrary state is represented by its projections onto each of the base states. The coffiecients are those projections.

- Warren

Tom Mattson
Staff Emeritus
Gold Member
Rev Prez said:
Wait a second, in the Schrodinger equation the amplitude is scaled by the Hamiltonian. The state vectors are already normalized,
Yes, I should have noted that the probability interpretation only follows if the state vector is normalized.

so what's the point of the coefficient?
The coefficients are needed because in general not all of those probabilities are equal. The coefficients can be thought of as weights for each basis state.

Okay, I can rephrase my question now.

I have an entangled ensemble of spatially separated and its Hamiltonian. Can I treat that Hamiltonian as I would the stress energy tensor and solve for it from a metric that defines a wormhole?

chroot
Staff Emeritus
Gold Member
The Hamiltonian is an operator with only two indices; the stress-energy tensor has four.

- Warren

chroot said:
The Hamiltonian is an operator with only two indices; the stress-energy tensor has four.

- Warren

All right, then. So how do you define the total energy of a state in a relativistic quantum field theory?

The following link suggests that entangled particles can be viewed as connected via nontrivial topologies - such as an Einstein-Rosen bridge - as a potential explanation for the Bell inequalities,

http://dabacon.org/pontiff/?p=869

One of the most interesting questions to ponder is what would Einstein’s reaction have been to Bell inequality violations by quantum theory. John Bell was able to show that correlations produced between spacelike separated quantum systems cannot in general be explained by local degrees of freedom carried with these systems. Reading the Einstein-Rosen paper, in which nontrivial topology is introducted without blinking, I’m inclined to think that Einstein would have thought of Bell’s result not as invalidating “classical” reasoning about quantum theory, but instead as a validation of the point of view advocated in this paper: that quantum theory is a consequence of a topological extension of general relativity.

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