# Wormholes and Quantum Entanglement

1. May 9, 2005

### Rev Prez

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?

2. May 9, 2005

### chroot

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

- Warren

3. May 9, 2005

### Rev Prez

I know. My question is can you generate a wormhole between two nonlocal, entangled states?

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

4. May 9, 2005

### chroot

Staff Emeritus
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

5. May 9, 2005

### Rev Prez

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?

Last edited: May 9, 2005
6. May 9, 2005

### chroot

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

- Warren

7. May 9, 2005

### Tom Mattson

Staff Emeritus
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 $ij$.

8. May 9, 2005

### Rev Prez

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

9. May 9, 2005

### Rev Prez

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?

10. May 9, 2005

### chroot

Staff Emeritus
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

11. May 9, 2005

### Tom Mattson

Staff Emeritus
Yes, I should have noted that the probability interpretation only follows if the state vector is normalized.

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.

12. May 10, 2005

### Rev Prez

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?

13. May 11, 2005

### chroot

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

- Warren

14. May 11, 2005

### Rev Prez

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

15. May 12, 2005

### christaltman

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

Last edited: May 12, 2005
16. May 12, 2005