# What does it mean to say that string theory is an example of "third quantisation"?

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I have read in a number of places, such as in some of the answers to Is a third quantisation possible? that String Theory is somehow an example of "third quantisation". Is this true? What exactly does this mean?

In quantum mechanics, say, the particle position $x^\mu$, becomes non-deterministic or an eigenvalue of an operator and instead, the wavefunction $\psi\left(x^\mu\right)$ is a deterministic complex scalar field which describes the probability amplitude of the particle's position. In Quantum Field Theory, the quantum field $\Phi\left(\psi\left(x^\mu\right)\right)$ is now deterministic and the wavefunction isn't.

Is the value of the quantum field itself non-deterministic in string theory? This really doesn't make sense to me; there certainly are field equations which can determine the values of the quantum fields.

Could this actually be related to the field-theory limit of string theory? Just like how locality, for example is actually preserved in (Perturbative) String Theory, but not in it's field theory limit?

It would be very nice if the downvoter could explain his downvote. What exactly is wrong with this question, or do you just disapprove of the terminology "Second Quantisation".

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The counting in "first, second, third" quantization only has a mathematical meaning - it quantifies a hierarchy of numbers, functions, functionals, functionalals, and so on that appear in the most straightforward description of the classical configuration at time $t$; or the wave function at time $t$.

Classical mechanics is "zeroth quantization" and it deals with numbers like $x,p$ - classical positions and momenta - at a given time. Quantum mechanics is obtained by promoting $x,p$ to operators, then $\psi(x)$, a function, expresses the state at a given moment.

Formally speaking, quantum field theory is approximately a result of the second quantization in which $\psi(x)$ is promoted to operators again - approximately field operators $\Phi(x,y,z)$ - and the state vector is given via functionals $\Psi[\Phi(x,y,z)]$. The number of numbers in this functional is "hierarchically higher" than it is for regular functions.

In string field theory, one formally goes one step further. The string fields $\Psi[x(\sigma)]$ which are themselves functionals, even in the classical (non-quantum) string field theory, are promoted to operators, so the state vector may be written as $\bar\Psi \{ \Psi[x(\sigma)] \}$, a functionalal.

In principle, you could imagine theories going even higher, to more complicated constructs. However, even the term "second quantization" is problematic because the wave function may also be written differently, without a functional - e.g. as a collection of functions of finitely many variables (wave functions for $n$ particles). That's just a result of a difference choice of a basis.

All physically relevant (which also includes the condition "separable") infinite-dimensional Hilbert spaces in quantum mechanical theories are isomorphic to each other, regardless whether they are obtained by third, second, or just first quantization. So all quantizations are physically really the same thing, qualitatively speaking, and the counting is just describing a heuristic way to quantify the number of numbers in the state vector that is however basis-dependent.

That's why even the "second quantization" term is outdated. Mathematically speaking, the QFT state vector may be written as a functional. But a QFT itself is a (first) quantization of a classical system, namely a classical field theory. It is not "another quantization" of the state vector $\psi(x,y,z)$ in non-relativistic quantum mechanics because this wave function is physically different from a classical field - they have very different interpretations even though, in some sense, they contain approximately "the same number of numbers".

To consider the fourth, fifth, or infinity-th quantization may be appealing but this identification is basis-dependent and one would still need some physical principles to pinpoint the interesting theories of the kind, so this method of getting more advanced theories probably isn't sufficient to do genuine progress in physics.

answered Apr 29, 2014 by (10,278 points)
Hi Lumo, thanks for these nice explanations and clarifications. I am very happy to see you here :-)

Thanks for this answer, and +1!

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