Spontaneous Symmetry Breaking and quantum mechanics

In summary, the conversation discusses the difficulty of understanding spontaneous symmetry breaking in simpler contexts, such as in quantum mechanics. It is noted that SSB is not possible in QM due to tunneling between different states, but becomes possible in QFT due to the transition probability becoming zero in infinite volume. Various textbooks are mentioned that touch on the topic of SSB, but none provide a complete approach. The discussion also touches on the connection between vacua and Goldstone's theorem, and the misconception that local gauge symmetries can be spontaneously broken. The book "Quantum Theory of Fields" by S. Weinberg is recommended as a clear and accurate source on the topic.
  • #1
arivero
Gold Member
3,459
154
TL;DR Summary
links and general commentary
Confronted with my inability to grasp Witten's Susy QM examples of supersymmetry breaking, I concluded that the problem was that I was not understanding spontaneous symmetry breaking in simpler contexts.
It seems that SSB is not possible in QM because of tunneling between the different states, and that it becomes possible in QFT because the transition probability becomes zero in infinite volume, albeit almost no textbook takes the time to prove it, just state it -or not-.
The best question I have found is this one in PhysicsExchange, and it links to an interesting paper couple of papers by Landsman
1687855907327.png
calculating the flea, they say in the cat instead of the elephant :cool: It also links to a preprint by Thirring and Narnhofer, that mentions the case of SUSY QM.

The double (and multiple) well symmetry breaking in QM was taken with great interest in the eighties, I guess the sparks were Witten's Susy QM, Coleman instantons, and the analysis of Jona-Lasinio, Martinelli and Scoppola, as well as Barry Simon's "flea in the elephant" for analysts. It was revisited periodically each time a generation finished their formative years. I see a revisit by M-K in 1992, by myself in 1994 (uploaded later), Casahorran in 2001 (triple well) or Alhendi-Lashin in 2004. Related topics as transparent or shape invariant potentials and delta potentials are also revisited frequently. Still I do not find a consistently complete approach to the topic, including things as Witten's spontaneous susy breaking, where the partner has been fully collapsed out and is not a normalised state anymore, or the impossibility of an analogous of Goldstone theorem for localization.
 
  • Like
Likes vanhees71
Physics news on Phys.org
  • #2
The simple picture is that in QFT you are dealing with infinite coupled copies of the QM double well problem, quasi an Ising problem. Hence if the overlap of the two wavefunctions in the two wells is x<1, in QFT the overlapp in QFT is ##\lim_{n \to \infty} x^n =0##. Hence there are two ground states which have no overlap and no operator which can only change the occupancy of a finite number of double wells, can have matrix elements between these two states. So the two ground states live in different Hilbertspaces.
Personally, I like the carefull anaysis by Strocchi: https://link.springer.com/book/10.1007/978-3-540-73593-9
 
  • Like
Likes dextercioby, arivero, topsquark and 1 other person
  • #3
arivero said:
QFT because the transition probability becomes zero in infinite volume
Source?
 
  • Like
Likes topsquark
  • #4
malawi_glenn said:
Source?
Good point, perhaps we could try to do a collaborative table telling the claims contained in each textbook. I will keep editing the table if new textbooks are commented in the thread.

BookSectionexcerpt or comments
Huang, "Quarks Leptons & Gauge Fields", 2nd edition3.2 Spontaneous Breaking of Global Gauge Invariancepage 53: "The transition amplitude between vacua with different values of [latex]\alpha_0[/latex] vanishes for infinite spatial volume."
Peskin Schroeder, "An Introduction to Quantum Field Theory"11.1 Spontaneous Symmetry Breaking.No discussion
11.3 to 11.5: the effective potentialpg 308 and figure 11.6 refer to metastable vacuum states, but only after the Effective Potential has been built. Volume is mentioned in eq. 11.50.
S. Weinberg, Quantum theory of Fields, vol. 2, Chpt. 19
Note that only global symmetries can be spontaneously broken, not local gauge symmetries, although even Weinberg talks about spontaneous breaking of local gauge symmetries. (@vanhees71 )
 
Last edited:
  • #5
Then why are checking for global minimum a thing in BSM Higgs physics modelleing? I.e. one puts theoretical restrictions on the parameter space of the model such that the potential does not have additional minima
 
  • #6
malawi_glenn said:
Then why are checking for global minimum a thing in BSM Higgs physics modelleing? I.e. one puts theoretical restrictions on the parameter space of the model such that the potential does not have additional minima
Not sure, Huang also puts an example of a negative quartic, that should be a metastable state in x=0 in quantum mechanics and says that "would be true in particle quantum mechanics, but not in quantum field theory" because in this case "the decay rate of such an initial state in infinite space is infinite" and thus the metastable case does not even exist. It would seem that authors have considered that decays rates of metastable vacuum states are an advanced topic and excused discusion in textbooks.

To put more wood in the fire, some other textbools like to use the connection between vacua as a way to "proof" Goldstone theorem. So Donoghue-Golowich-Holstein pg 21: "Thus in the limit of infinite wavelength the excitation energy vanishes, yielding a Goldstone boson"
 
  • #7
I can ask a former collegue who is working on similar research problems
 
  • #8
malawi_glenn said:
I can ask a former collegue who is working on similar research problems
It is a bit of a labyrinth. Literally: Peskin-Schroeder say that other exotic effects come from "the topology" of the space of vacua. Once you leave the mexican hat, everything can happen, it seems.
 
  • Like
Likes vanhees71 and malawi_glenn
  • #9
I think the clearest discussion of spontaneous symmetry breaking is in

S. Weinberg, Quantum theory of Fields, vol. 2, Chpt. 19

Note that only global symmetries can be spontaneously broken, not local gauge symmetries, although even Weinberg talks about spontaneous breaking of local gauge symmetries. It's an unfortunate inaccuracy of language. One should rather talk about a "Higgsed local gauge symmetry" or "hidden local gauge symmetry", but that's another topic.
 
  • Like
Likes arivero
  • #10
Weinbergs Discussion is certainly correct. But him stressing the importance of degenerate vacua is misleading as in QFT vacua are generally highly degenerate.
In the Weinberg chair example, e.g., vaporization of the chair would lead to a gas with some given density. If we could switch off the interactions between the gas particles, this gas would form a vacuum state with unbroken rotational symmetry. However, in the thermodynamic limit, total angular momentum is not an observable, only angular momentum density is. We could change the angular momentum of an arbitrary (finite) number of gas particles without changing the state. Hence the ground state is degenerate even if symmetry is unbroken, but the structure of the ground states is different.
For the example of superconductivity this has been worked out very lucidly by Rudolf Haag, which is my favourite reference on the topic of broken symmetry:

Haag, R. (1962). The mathematical structure of the Bardeen-Cooper-Schrieffer model. Il Nuovo Cimento (1955-1965), 25(2), 287-299.
 
  • Like
Likes dextercioby and vanhees71
  • #11
Another very pedagogical paper using the BCS model to discuss the important difference between spontaneous symmetry breaking and hidden local gauge symmetries is

https://arxiv.org/abs/cond-mat/0503400
 
  • Like
Likes dextercioby
  • #12
When Hendrik mentioned this paper before, I tended to start arguing, especially about section II. The rest of the paper is excellent. But maybe I finally see what I got wrong. I'll start a new thread to discuss this.
 
  • Like
Likes arivero, vanhees71 and dextercioby
  • #13
arivero said:
It seems that SSB is not possible in QM
There is a lot of SUSY for ordinary QM, and in all interesting cases, the SUSY is broken. See, e.g.,

 
  • Like
Likes vanhees71
  • #14
A. Neumaier said:
There is a lot of SUSY for ordinary QM, and in all interesting cases, the SUSY is broken. See, e.g.,

Ah thanks for the reminiscence, I did some calculations for https://inspirehep.net/literature/355497 a preprint then. I guess I should check it again. It was a line of research starting from https://inspirehep.net/literature/18852 in the time that Boya was going back and forward to utexas, but I was never sure if it was susy for real or just factorisation method.
 
  • #15
arivero said:
Ah thanks for the reminiscence, I did some calculations for https://inspirehep.net/literature/355497 a preprint then. I guess I should check it again. It was a line of research starting from https://inspirehep.net/literature/18852 in the time that Boya was going back and forward to utexas, but I was never sure if it was susy for real or just factorisation method.
The factorization method for exactly solvable QM systems can be interpreted in N=1 SUSY terms. This shows that SUSY is nothing spectacularly new.
 
  • Like
Likes vanhees71 and arivero
  • #16
Boya was particularly intrigued by the case where the superpotential is the sign function, so the two pairs are delta potential and delta barrier. I was assigned to look into it but I was never explicitly told to focus in susy pairings so I let it to pass. Surely it is done elsewhere in the literature multiple times; I find this kind of things resurface with a ten years periodicity.
 
  • Like
Likes vanhees71

Similar threads

  • High Energy, Nuclear, Particle Physics
Replies
34
Views
3K
  • Quantum Physics
3
Replies
75
Views
8K
Replies
4
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
3K
  • Atomic and Condensed Matter
Replies
2
Views
2K
Replies
7
Views
3K
Replies
3
Views
4K
  • Beyond the Standard Models
Replies
4
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
4K
Back
Top