Representations in Quantum Physics

In summary, we discussed the confusion that can arise when discussing representations in physics, particularly in the context of group representations and representations of Lie algebras. We also touched on the importance of irreducible representations and their connection to elementary particles. Additionally, we clarified that in physics, we are typically dealing with infinite-dimensional Hilbert spaces due to the need for unitary representations of the Poincare group. Finally, we examined the concept of "generators", which can refer to Hermitian operators that correspond to observables such as total momentum and angular momentum.
  • #1
fresh_42
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I'm not asking about the math here, I'm interested in the wording physicists use in QP / QM / QFT. I'm frequently confused, when I'm reading threads here. They often start completely underdetermined and often also just wrong from a mathematical point of view, but seemingly, physicists know what it is about.

  • If someone speaks of "representation of G" where G is likewise a simple Lie group as e.g. SU(2) or SU(5), or what I've just read the Poincaré group, or Heisenberg group, do they really mean group representations, or the representations of the corresponding Lie algebras? How is it meant? As said, or as a very questionable use of language here? I often read about group representations, but the equations which sometimes follow are those of Lie algebras.
  • Now to the representations of the Lie algebras. Why irreducible? Is it simply because of their minimality, i.e. that all others are sums of them, or is there something with a physical meaning, that irreducible representations have and others don't have? Also, which scalar field can be assumed here? Is there always implicitly meant ##\mathbb{C}## to avoid trouble with weights and eigen values, or at least ##\operatorname{char} \mathbb{F} =0##, because basically nobody feels the need to mention it? I mean at least real or complex should be mentioned, from a mathematical point of view. What is it, that physicists don't feel the need to say what they mean?
  • And finally, what about infinite dimensional representations? As soon as someone says quantum, Hilbert spaces come around the corner, so where did they go to, when it comes to representations?
 
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  • #2
I am going to tell you about Poincare group representations, because this group is physically the most transparent and relevant. The importance of other groups, like SU(5), is rather hypothetical.
  • We are interested in representations of both the group itself and its Lie algebra. Both representations have their significance in physics. The group representation will tell you how states and observables transform with respect to inertial transformations of observers. Generators of the Lie algebra representation are identified with basic observables (total momentum, total energy, total angular momentum, boost). Their Lie brackets translate directly into quantum commutators of these observables. Representations of the group and its Lie algebra are closely related by an exponential mapping, so physicists sometimes confuse the two.
  • Irreducible representations of the Poincare group are special, because they correspond to systems without internal structure = elementary particles. Systems of two or more particles are described by reducible representations. Roughly speaking, these systems have non-trivial mass spectra, and each mass eigenvalue corresponds to one irreducible summand. Postulates of quantum mechanics permit real, complex or quaternionic Hilbert spaces. There are some works on R- and Q- quantum mechanics, but they are rather exotic. So, in 99.9% of cases we are dealing with group representations in the complex Hilbert space.
  • Even single particle has infinite-dimensional Hilbert space (= representation space of an irreducible representation of the Poincare group). So, strictly speaking, we are always dealing with infinite-dimensional Hilbert spaces in physics. When people are studying finite-dimensional representations (e.g., spin states and the rotation group), they are usually focusing on a slice of the total Hilbert space, e.g., assuming that the total momentum is zero.
Eugene.
 
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  • #3
Thanks for the answer, Eugene. Really, no kidding. Nevertheless I have still a few questions and remarks.
meopemuk said:
We are interested in representations of both the group itself and its Lie algebra.
Sounds like a good reason to distinguish them and not just say representation and let the reader guess which one. And it leaves me as clueless as before. Not to mention this stupid term "generator". I would really like to know, who introduced this term, when and even more: why. It sounds like a small engine sitting on a curve through the group, generating a tangential motion. Funny, but only partially accurate.
meopemuk said:
Representations of the group and its Lie algebra are closely related by an exponential mapping
This is true for the adjoint representation. Does it hold for any? I don't remember having read that for arbitrary representations, but I'm not sure. And, yes, I've forgotten to mention, that often people speak about "(the) representation" if they mean the adjoint representation only. And this can usually only be found out by the eigenvalues / weights they consider. Unfortunately both, the one of the group as of the algebra are called adjoint, so no help here to figure out what is meant. It drives me mad: group or algebra? adjoint or any? finite or infinite dimensional? An algebraically closed scalar field of characteristic zero makes sense, so I don't want to complain about this. It's just that the entire representation theory seems to consist of merely a tiny part of it: ##\operatorname{ad}_\mathbb{C}\mathfrak{su}(n)## for ##n=1,2,3##.
meopemuk said:
So, strictly speaking, we are always dealing with infinite-dimensional Hilbert spaces in physics.
Not really. Usually there are questions about the adjoint representation(s) - finite dimensional, or the weight spaces of irreducible representations of the SU gauge groups - finite dimensional. There is mostly also a natural representation space as given by the action of the operators, which could be infinite dimensional, e.g. for differential operators, but these are usually not meant.
 
  • #4
I forgot to mention that in physics we are mostly interested in unitary representations of the Poincare group, because any change of observer is supposed to preserve quantum probabilities. This is the reason why representation Hilbert spaces are always infinite-dimensional (the non-compact Poincare group cannot have finite-dimensional unitary representations).

Regarding "generators", here is one example. Translations in space form a subgroup of the Poincare group. Let ##U_{\mathbf{a}}## be a unitary operator representing translation by the vector ##\mathbf{a}##. We can write this operator in the exponential form: ##U_{\mathbf{a}}= e^{-\frac{i}{\hbar}\mathbf{P}\cdot \mathbf{a}}##, where ##\mathbf{P}## is the Hermitian operator which is called the "generator of translations" or the "operator of total momentum". There are 7 other Hermitian generators, corresponding to rotations, time translations and boosts. Their physical interpretation is as observables of the total angular momentum, energy (Hamiltonian) and boost. These 10 Hermitian operators satisfy the commutation relations of the Poincare Lie algebra.

You can read more in S. Weinberg's "The quantum theory of fields", vol. 1.

Eugene.
 
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  • #5
fresh_42 said:
This is true for the adjoint representation. Does it hold for any?
If you have a representation of a Lie group, its generators always form a representation of the corresponding Lie algebra. On the other hand, if a Lie group is connected and simply connected, then a representation of its Lie algebra always integrates to a representation of the Lie group. Physicists usually consider all representations, not just the adjoint representation. However, some representations may be physically excluded, such as the negative m^2 reps of the Poincare group for example.

Not really. Usually there are questions about the adjoint representation(s) - finite dimensional, or the weight spaces of irreducible representations of the SU gauge groups - finite dimensional. There is mostly also a natural representation space as given by the action of the operators, which could be infinite dimensional, e.g. for differential operators, but these are usually not meant.
No, usually, we are indeed interested in infinite-dimensional representations. Finite-dimensional representations are usually relevant only for toy models and simplified systems. (There are exceptions.)
 
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  • #6
Well, physicists are pretty sloppy from the point of view of mathematicians (not always to the good for the physicists to say it carefully ;-)). Indeed, in quantum physics you come very far by considering only the representations of the Lie algebras, because after all the whole dynamics in QT is in the commutation relations among the (usually essentially self-adjoint) operators representing observables and states.

It's also important to know that a priori not the unitary representations of (symmetry) groups are relevant but the socalled "unitary ray representations" which often induce unique unitary representations (e.g., Poincare group, whose Lie algebra has no non-trivial central charges) or are substituted by unitary representations of central extensions of the original group/algebra (Galileo group, whose unitary representations do not lead to any useful dynamics, while standard QM provides a ray representation of this group or a unitary representation of a central extension of the Galileo group with the mass operator as a central charge of the corresponding Lie algebra).

To look for the irreducible representations first makes sense since you can decompose any reducible one in its irreducible parts which evolve dynamically independently from each other.
 
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  • #7
vanhees71 said:
In may cases, such ray representations can be "lifted" to unitary representations, i.e., you can redefine the phase factors of the ##U(g_j)## such that for all ##g_1## and ##g_2## the phase ##\phi(g_1,g_2)=0##. This is investigated in detail by Weinberg, and it's true for the proper orthochronous Poincare group, but it's not the case for the Galileo group in non-relativistic quantum mechanics, because the true unitary representations of the Galileo group do not lead to physically interpretible quantum theoretical description of any known object in nature. The classical Galileo group can be extended to a quantum Galileo group with the mass as a socalled central charge of the corresponding Galileo Lie algebra. In contradistinction mass in the case of the proper orthochronous Poincare group mass (squared) is a Casimir operator, and this explains the subtle differences between the concept of mass in Newtonian and special-relativistic physics.

vanhees71 said:
It's also important to know that a priori not the unitary representations of (symmetry) groups are relevant but the socalled "unitary ray representations" which often induce unique unitary representations (e.g., Poincare group, whose Lie algebra has no non-trivial central charges) or are substituted by unitary representations of central extensions of the original group/algebra (Galileo group, whose unitary representations do not lead to any useful dynamics, while standard QM provides a ray representation of this group or a unitary representation of a central extension of the Galileo group with the mass operator as a central charge of the corresponding Lie algebra).

Could you (@vanhees71) or anyone else kindly point to some text(book) with a physicist-friendly introduction to those aspects or Lie groups/algebras that are mentioned above (i.e., definition of the central charge and Casimir operators as well as the application to the Poincaré and Galilei group)?
Thanks!
 
  • #8
odietrich said:
point to some text(book) with a physicist-friendly introduction to those aspects or Lie groups/algebras that are mentioned above (i.e., definition of the central charge and Casimir operators as well as the application to the Poincaré and Galilei group)?

It is difficult to find one book that fits these criteria. "Theory of Group Representations and Applications" by Barut and Raczka has all the technical stuff that you mentioned (and much more, maybe too much more), but, even though it is written by physicists, I am not sure that it is a "physicist-friendly introduction". Perhaps other folks are familiar with this book and can comment. Also, it is quite long.

https://www.amazon.com/dp/9971502178/?tag=pfamazon01-20
 
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  • #9
George Jones said:
It is difficult to find one book that fits these criteria. "Theory of Group Representations and Applications" by Barut and Raczka has all the technical stuff that you mentioned (and much more, maybe too much more), but, even though it is written by physicists, I am not sure that it is a "physicist-friendly introduction". Perhaps other folks are familiar with this book and can comment. Also, it is quite long.

https://www.amazon.com/dp/9971502178/?tag=pfamazon01-20
I think it is a good book. Not sure if physicists would like it.
 
  • #10
odietrich said:
Could you (@vanhees71) or anyone else kindly point to some text(book) with a physicist-friendly introduction to those aspects or Lie groups/algebras that are mentioned above (i.e., definition of the central charge and Casimir operators as well as the application to the Poincaré and Galilei group)?
Thanks!
Zee is very physicist-friendly
https://www.amazon.com/dp/0691162697/?tag=pfamazon01-20
 
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  • #11
fresh_42 said:
They often start completely underdetermined and often also just wrong from a mathematical point of view, but seemingly, physicists know what it is about.
This is the case for most of theoretical physics, unless done by mathematical physicists. Only the latter are trained to mathematical standards of rigor; the others learn math like a language, to be picked up and used as they see and hear it being done by other physicists.
fresh_42 said:
  • If someone speaks of "representation of G" where G is likewise a simple Lie group as e.g. SU(2) or SU(5), or what I've just read the Poincaré group, or Heisenberg group, do they really mean group representations, or the representations of the corresponding Lie algebras? How is it meant? As said, or as a very questionable use of language here? I often read about group representations, but the equations which sometimes follow are those of Lie algebras.
  • Also, which scalar field can be assumed here? Is there always implicitly meant ##\mathbb{C}## to avoid trouble with weights and eigenvalues, or at least ##\operatorname{char} \mathbb{F} =0##, because basically nobody feels the need to mention it? I mean at least real or complex should be mentioned, from a mathematical point of view. What is it, that physicists don't feel the need to say what they mean?
In quantum physics, a representation usually means in precise mathematical terms a continuous unitary projective representation over the field of complex numbers. This field is the only one relevant in quantum physics, and hence it is everywhere in QM understood implicitly. Similarly, unitarity is basic for quantum mechnaics, and in the few cases where nonunitary representations are relevant, the lack of unitarity is usually spelled out explicitly.

For finite-dimensional Lie groups, every continuous group representation gives rise to a corresponding Lie algebra representation, and conversely. Therefore physicists think of the two as being given simultaneously and they freely move between them. In infinite dimensions there are conditions that must hold in order that this equivalence is still valid, but (like with interchanges of limits, integrals, and derivatives) physicists assume by default that everything is right unless they are forced to look at the technical details by arriving at nonsense.
fresh_42 said:
  • Now to the representations of the Lie algebras. Why irreducible? Is it simply because of their minimality, i.e. that all others are sums of them, or is there something with a physical meaning, that irreducible representations have and others don't have?
Irreducible representations contain the essence, as any other representation can (in the unitary case) be reduced to a direct sum or integral of irreducible representations.
fresh_42 said:
  • And finally, what about infinite dimensional representations? As soon as someone says quantum, Hilbert spaces come around the corner, so where did they go to, when it comes to representations?
Already the continuous unitary representations of the canonical commutation relations (defining the Heisenberg Lie algebra and group) are infinite-dimensional. Nontrivial unitary representations are finite-dimensional only for compact groups. But the latter case is important: For example rotational symmetry manifests itself in quantum mechanics through the representation theory of the compact group SO(3) of rotations in 3 dimensions, which only has finite-dimensional representations. The irreducible ones are classified by spin, the reducible ones obtained by taking tensor products lead to Clebsch-Gordan formulas etc.
 
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  • #12
odietrich said:
Could you (@vanhees71) or anyone else kindly point to some text(book) with a physicist-friendly introduction to those aspects or Lie groups/algebras that are mentioned above (i.e., definition of the central charge and Casimir operators as well as the application to the Poincaré and Galilei group)?
Thanks!

I like the physicist-friendly presentation given in

D. G. Currie, T. F. Jordan and E. C. G. Sudarshan, "Relativistic invariance and Hamiltonian theories of interacting particles", Rev. Mod. Phys., 35 (1963), 350.

see, especially, their Appendix B. You can also try

V. S. Varadarajan, "Geometry of quantum theory", 2nd edition, Springer 2007.

Eugene.
 
  • #14
Thanks for your suggestions! I'm currently trying to find out how to learn from these about Casimir operators as well as the central charge (is this the same as the central extension defined by Barut and Raczka?) and the Galilei group. A first quick search yielded nothing about central charges or Casimir operators in the texts by Currie et al. (1963) or Varadarajan (2007). Zee gives some details around p.510. Barut and Raczka might provide more information, but are certainly less accessible - I'll need some more time time for that.
 
  • #15
odietrich said:
Could you (@vanhees71) or anyone else kindly point to some text(book) with a physicist-friendly introduction to those aspects or Lie groups/algebras that are mentioned above (i.e., definition of the central charge and Casimir operators as well as the application to the Poincaré and Galilei group)?
Thanks!
S. Weinberg, The Quantum Theory of Fields Vol. 1, Cambridge University Press

discusses this in great detail. It's not an easy read, but the time spent to understand it is well spent time for sure! For a detailed coverage of the Newtonian/Galilean case, see

L. Ballentine, Quantum Mechanics - A modern development, World Scientific
 
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  • #16
odietrich said:
Thanks for your suggestions! I'm currently trying to find out how to learn from these about Casimir operators as well as the central charge (is this the same as the central extension defined by Barut and Raczka?) and the Galilei group. A first quick search yielded nothing about central charges or Casimir operators in the texts by Currie et al. (1963) or Varadarajan (2007). Zee gives some details around p.510. Barut and Raczka might provide more information, but are certainly less accessible - I'll need some more time time for that.
Maybe you like the discussion about central charges in my thesis,

https://www.rug.nl/research/portal/...ed(fb063f36-42dc-4529-a070-9c801238689a).html

The Bargmann algebra already pops up at the classical level for point particles.
 
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1. What are representations in quantum physics?

Representations in quantum physics refer to the mathematical framework used to describe the behavior of particles and systems on a quantum level. They involve the use of abstract mathematical objects, such as matrices and vectors, to represent physical quantities and states.

2. Why are representations important in quantum physics?

Representations are important in quantum physics because they allow us to mathematically describe and analyze the behavior and properties of quantum systems. They provide a bridge between the abstract mathematical formalism of quantum mechanics and the observable physical world.

3. How do representations differ from classical physics?

Representations in quantum physics differ from those in classical physics in that they involve the use of complex numbers, non-commutative operations, and the concept of superposition. This is due to the probabilistic nature of quantum mechanics and the uncertainty principle.

4. What are some common representations used in quantum physics?

Some common representations used in quantum physics include the Schrödinger representation, Heisenberg representation, and Dirac representation. These representations use different mathematical formulations to describe quantum systems and are useful for different types of calculations and analyses.

5. How do representations impact our understanding of quantum phenomena?

Representations play a crucial role in our understanding of quantum phenomena by providing a framework for analyzing and interpreting experimental results. They allow us to make predictions and explain the behavior of particles and systems on a quantum level, leading to a deeper understanding of the fundamental laws of nature.

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