No simple map between classical and quantum

  • Thread starter Lapidus
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In Sean Carroll's GR book I found the following statement:

there is no simple map between classical and quantum theories,
- there are classical theories with no quantum counterpart
- classical theories with multiple quantum versions
- quantum theories without any classical analogue

Could someone give me examples to each of the three cases?

thanks
 

Avodyne

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"classical theories with no quantum counterpart"

I don't know of any example.

"classical theories with multiple quantum versions"

EVERY classical theory has multiple quantum versions, because, given a classical observable F(q,p), you have to decide what its quantum counterpart is, and in general you have an operator-ordering ambiguity.

"quantum theories without any classical analogue"

Spin 1/2.
 
343
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Thanks for answering!

I took the statement from page 380 of the book "Spacetime and geometry" by Sean Carroll.
 

dextercioby

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"classical theories with no quantum counterpart"

I don't know of any example.
General relativity.
 

Avodyne

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General relativity most certainly does have a quantum counterpart. It's relatively (no pun intended!) straightforward to canonically quantize it; the problem is that you end up with a nonrenormalizable quantum field theory with an infinite number of parameters that must be specified. Since both GR and QM are part of nature, it is virtually (no pun intended!) certain that there is some "ultraviolet completion" of canonically quantized GR. It is widely but not universally (no pun intended!) believed among experts that string theory provides such an ultraviolet completion.

So it's hard to believe that this is what Sean Carroll had in mind ...
 

tom.stoer

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- there are classical theories with no quantum counterpart
- classical theories with multiple quantum versions
- quantum theories without any classical analogue
- theories with quantization anomalies (e.g. certain chiral gauge theories)
- most classical theories become ambiguous during quantization due to operator ordering
- ?
 

DrDu

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There are also theories which comprise both classical and quantum variables at the same time. E.g. Temperature is a classical parameter which nevertheless appears in quantum field theories together with quantized field variables.
 

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