No simple map between classical and quantum

In summary, Sean Carroll discusses the lack of a simple mapping between classical and quantum theories in his book "Spacetime and Geometry." This topic is further explored in a thread on Physics Forums, where examples are given for classical theories with no quantum counterpart, classical theories with multiple quantum versions, and quantum theories without any classical analogue. Some examples mentioned are spin 1/2, general relativity, theories with quantization anomalies, and theories with both classical and quantum variables.
  • #1
Lapidus
344
11
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
 
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  • #3
"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.
 
  • #4
Thanks for answering!

I took the statement from page 380 of the book "Spacetime and geometry" by Sean Carroll.
 
  • #5
Avodyne said:
"classical theories with no quantum counterpart"

I don't know of any example.

General relativity.
 
  • #6
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 ...
 
  • #7
Lapidus said:
- 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
- ?
 
  • #8
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.
 

1. What is the difference between classical and quantum physics?

Classical physics is the branch of physics that studies the behavior of macroscopic objects and systems, while quantum physics deals with the behavior of subatomic particles and their interactions.

2. Is there a simple map between classical and quantum physics?

No, there is no simple map between classical and quantum physics. The two theories have fundamentally different principles and mathematical frameworks, making it difficult to directly translate between them.

3. Can classical and quantum theories be unified?

There have been attempts to unify classical and quantum theories, such as in the field of quantum gravity, but so far no complete and satisfactory theory has been established.

4. How does the uncertainty principle play a role in the relationship between classical and quantum physics?

The uncertainty principle, which states that the position and momentum of a particle cannot be known simultaneously, is a key difference between classical and quantum physics. In classical physics, the position and momentum of a particle can be precisely measured, while in quantum physics, there is always some level of uncertainty in these measurements.

5. Are there any real-world applications of the relationship between classical and quantum physics?

The relationship between classical and quantum physics has led to many technological advancements, such as the development of transistors, lasers, and other devices that rely on quantum principles. Additionally, understanding the relationship between the two theories is crucial for further advancements in fields such as quantum computing and quantum cryptography.

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