# B Is string theory a deterministic hidden variable theory?

#### Aidyan

There are several interpretations of QM which differ from being deterministic or non deterministic, with or without hidden variables and local or non-local. As I understand it ST poses the existence of vibrating strings, that is, physical objects with definite properties, moving along a world sheet coordinate system having determinate position and momentum and where interactions and correlations are expresses as local string dynamics. Does this imply that it is a deterministic and local theory? And where form comes quantum randomness in ST? Is it a consequence of the strings vibration? If so, could one then say that it is a hidden variable theory?

#### atyy

String theory is a hidden variable theory (strings), but it is a quantum hidden variable theory, so it does not have much to say beyond standard quantum theory about about the issues of locality/nonlocality and determinism/randomness involved in violation of a Bell inequality.

There does seem to be another form of nonlocality in string theory related to the Holographic principle. Here is a Scientific American article about the holographic principle in string theory by Juan Maldacena, who first concretely raised the conjecture: https://www.scientificamerican.com/article/the-illusion-of-gravity-2007-04/

#### Demystifier

2018 Award
Classical string theory is local and deterministic. But quantum string theory is no more local and deterministic than other quantum theories. The statement that string "vibrates" usually refers to classical (not quantum) strings. Hence string theory, in its usual form, is not a hidden variable theory, in the sense that it does not explain where does quantum randomness come from. But it is possible to introduce hidden variables for string theory, just as it possible to do it for any other quantum theory.

#### haushofer

Just to add: non-locality is a problem when you want to quantize branes. The reason is intuitively as follows: stretchin a string costs energy, so strings are localized in space. But branes, e.g. 2 branes, can be stretched indefinitely without energy cost. Consider for example a long peak on a 2 brane, which you make arbitrarily narrow. As such you can make the corresponding area (and hence energy!) arbitrarily small. With this, the brane can "peak out" arbitrarily far without energy cost. As such the quantization is troublesome. To be honest, I'm not sure to which extent the quantization of branes is well-defined nowadays, but I like this intuitive picture.

#### MathematicalPhysicist

Gold Member
Classical string theory is local and deterministic. But quantum string theory is no more local and deterministic than other quantum theories. The statement that string "vibrates" usually refers to classical (not quantum) strings. Hence string theory, in its usual form, is not a hidden variable theory, in the sense that it does not explain where does quantum randomness come from. But it is possible to introduce hidden variables for string theory, just as it possible to do it for any other quantum theory.
I find it hard to grasp a theory such as string theory called either Classical or Quantum, where an alleged theory of QGR should be neither quantum nor classical; obviously in the appropriate limits we should get back both theories, but in the intermediate we should get something different from both, otherwise it's not a different theory but a patch of differing theories.

Now, this is akin the question what are those limits?

#### stevendaryl

Staff Emeritus
Classical string theory is local and deterministic. But quantum string theory is no more local and deterministic than other quantum theories. The statement that string "vibrates" usually refers to classical (not quantum) strings. Hence string theory, in its usual form, is not a hidden variable theory, in the sense that it does not explain where does quantum randomness come from. But it is possible to introduce hidden variables for string theory, just as it possible to do it for any other quantum theory.
I had the idea that string theory was nonlocal even classically, although the nonlocality is only significant over short distances.

https://arxiv.org/pdf/1310.4957.pdf

#### PAllen

I find it hard to grasp a theory such as string theory called either Classical or Quantum, where an alleged theory of QGR should be neither quantum nor classical; obviously in the appropriate limits we should get back both theories, but in the intermediate we should get something different from both, otherwise it's not a different theory but a patch of differing theories.

Now, this is akin the question what are those limits?
I don't think it is a given that the QG solution is neither quantum nor classical. For example, asymptotic safety would be a pure quantum approach that would (if it worked) fully solve quantum gravity and include a complete theory of matter (unlike e.g. LQG, which in addition to other issues, has not yet accommodated anything like the standard model). That AS (asymptotic safety) does nothing to resolve various other issues in physics (the observations bundled in "the dark matter problem", the cosmological constant problem, etc.) is irrelevant. There is no a priori expectation that a QG solution should resolve these.

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#### Demystifier

2018 Award
an alleged theory of QGR should be neither quantum nor classical
Where did you get it from?

2018 Award

#### MathematicalPhysicist

Gold Member
@Demystifier I assumed that just like GR isn't Newtonian Gravity and QM isn't CM, but in the appropriate limits they coincide, i.e. when $\hbar \to 0$ QM becomes CM, and GR becomes NG in an appropriate limit which I am not sure what it is, then also QGR should obey the same behaviour, otherwise why call it a different theory than QM or GR?

#### Demystifier

2018 Award
@Demystifier I assumed that just like GR isn't Newtonian Gravity and QM isn't CM, but in the appropriate limits they coincide, i.e. when $\hbar \to 0$ QM becomes CM, and GR becomes NG in an appropriate limit which I am not sure what it is, then also QGR should obey the same behaviour, otherwise why call it a different theory than QM or GR?
Just because a theory is not quantum mechanics doesn't mean that this theory isn't quantum.

#### MathematicalPhysicist

Gold Member
Just because a theory is not quantum mechanics doesn't mean that this theory isn't quantum.
Then what makes a theory quantum?

I have seen the trichotomy in Zelevinsky's textbook of Quantum Mechanics, Quantum Physics and another term which I have forgotten.
Shouldn't a new theory transcend beyond the quantum?

#### Demystifier

2018 Award
Then what makes a theory quantum?

I have seen the trichotomy in Zelevinsky's textbook of Quantum Mechanics, Quantum Physics and another term which I have forgotten.
Shouldn't a new theory transcend beyond the quantum?
For instance, quantum field theory is not quantum mechanics, but is quantum. Quantum mechanics is a quantum theory of pointlike particles.

#### MathematicalPhysicist

Gold Member
For instance, quantum field theory is not quantum mechanics, but is quantum. Quantum mechanics is a quantum theory of pointlike particles.
OK, so string theory is quantum of strings which aren't fields nor points.

So the thing that makes something 'quantum' is what exactly?

#### MathematicalPhysicist

Gold Member
Obviously I know the maths of QM and QFT, but what makes these two theories quantum I am not sure.
Maybe because no technical book that I read discuss this matter.

#### Demystifier

2018 Award
So the thing that makes something 'quantum' is what exactly?
If observables are represented by non-commuting operators, then it's quantum. If probabilities are given by something of the form $|\psi|^2$, then it's quantum.

#### MathematicalPhysicist

Gold Member
If observables are represented by non-commuting operators, then it's quantum. If probabilities are given by something of the form $|\psi|^2$, then it's quantum.
In QM $\psi$ are wave-functions, in QFT they are wave-functionals.
So only these two conditions are enough to declare a theory as quantum.

#### Demystifier

2018 Award
In QM $\psi$ are wave-functions, in QFT they are wave-functionals.
So only these two conditions are enough to declare a theory as quantum.
Superficially speaking, yes. But if you want a deeper insight, see my paper https://arxiv.org/abs/quant-ph/0505143 where classical mechanics is represented by a quantum-like formalism.

#### stevendaryl

Staff Emeritus
Are you one of the authors of that paper?
No! Did I give that impression, somehow?

I actually was trying to find a John Baez comment from years ago saying that string theory was slightly nonlocal. I say "slightly", because the nonlocality was confined to a region the size of the length of the string. I didn't remember the argument for why.

#### stevendaryl

Staff Emeritus
OK, so string theory is quantum of strings which aren't fields nor points.

So the thing that makes something 'quantum' is what exactly?
The difference between a classical field theory and a quantum field theory is that whereas a classical field theory is deterministic--the configuration of the field and its sources at one time determines the configuration at later times---a quantum field theory is nondeterministic. The equations of quantum field theory describe not the fields themselves but probability amplitudes for the field.

The relationship between classical field theory and quantum field theory is very similar to the relationship between Newtonian physics and the Heisenberg equations of motion for quantum mechanics. The Newtonian equations of motion (or actually, the Euler-Lagrange equations, which are equivalent) for the harmonic oscillator, say, are:

$\frac{dx}{dt} = \frac{p}{m}$
$\frac{dp}{dt} = -K x$

Those are exactly the same as the Heisenberg equations of motion for a harmonic oscillator, but in the Heisenberg equations, $x$ and $p$ are operators, rather than real-valued functions of time.

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