# What is 3rd quantization and what can be learned from it?

## Main Question or Discussion Point

I've heard that the quantum mechanics of particles in a potential is called 1st quantization which produces wavefunctions. And I've heard that quantum field theory is called 2nd quantization which in turn quantizes the wavefunctions. Can this process be iterated to give 3rd quantization, and what would that tell us?

As I understand it, 1st quantization gives distribution functions for variables such as position and momentum. And 2nd quantization uses the functions of 1st quantization to get distribution functions for fields that can be used to calculate the excitation modes of fields that are interpreted as particles. Would 3rd quantization tell us which fields are allowed to exist in the first place that are then used in 2nd quantization? If so, then it seems to me that a 3rd quantization procedure might give us a relationship between QFT fields so that if we measured a property of one kind of field (say the EM field), it would automatically give us the properties of other kinds of fields (say the Strong Force) by means of this relationship between the kinds of fields that are allowed to exist. Any literature on this out there? Thanks.

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Ben Niehoff
Gold Member
3rd quantization is one way to think of what is happening in string field theory, although no one calls it that. In SFT, strings are quanta of a string field. The various modes of a string correspond to various spacetime fields, whose quanta are particles. And these particles have quantized position, momentum, etc.

String field theory can be said to be "3rd quantized", whereas string theory is "2nd quantized".

ie. while the quantization of string theory (which leads to different gauge and matter field) proceeds very similar to quantization in point quantum mechanics, string field theory is similar to point QFT.

What I guess I'm looking for is a summary of a natural progression from 1st to 2nd to 3rd quantization procedure - how the output of one is the input of the other - and how one is constrained by the higher order procedure. Is there anything like that in the literature? Thanks.

See the informal musings from Baez: http://math.ucr.edu/home/baez/nth_quantization.html

/Fredrik
I read that, thank you. It was too abstract for me. He made a number of statements about how to interpret the math that I am unable to evaluate. I'm left having to take his word for it, and I'm really not comfortable with that. He seems to think that string theory is 3rd quantization. But I have to wonder what the loop quantum gravity people would have to say about that. It would be easier for me if someone could show me the formulas in each of the 1st, 2nd, and 3rd quantization procedures, identify the variables in each, show how the output of one is being used as the input of the other, and explain what information is gained in each.

Fra
I'm on my way out in a moment so I can't expand, but IMO the best way to understand this is to stop thinking in terms of "particles" and other "mechanical pictures". The way I see it is that an a more abstract information picture, quantization is like an induction step. But there is a good reason (the formalization isn't fleshed out though) why this does not yield an infinite tower of turtles, and that IMHO has to do with limiting information capacity. All we see is a "window" of this tower. One step classical-QM is the smallest window of course. In 2nd quantization it's a bigger window.