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Ex particle theorist here. I think this answer is a bit misleading. Symmetries in quantum mechanics do typically act via $e^{iHt}$, but this isn't why amplitud

Ex particle theorist here. I think this answer is a bit misleading. Symmetries in quantum mechanics do typically act via $e^{iHt}$, but this isn't why amplitudes are complex. It's a consequence of the fact that symmetries generally act via unitary transformations, but to even speak of unitary transformations, one must already have amplitudes which are complex. A secondary complaint. None of the groups you mention play a central role in the machinery of quantum theory. They're important in some of the quantum mechanical models we use for particle physics. Arguably more important is the group R of translations in time. But even this isn't necessarily central. Quantum theory works just fine on curved spacetimes where time translation isn't a symmetry.

calhoun137 8y ago
Great points, let me try and explain better where the group O(1) appears in basic QM. An amplitude in QM is only defined up to a global phase transformation e^it. To be more clear, given some amplitude A and a real number t, we will calculate the same probability for any measurement by using the amplitude Ae^it instead. Well there is the group O(1) right there. What's super interesting is that this fact is literally the reason why charge is conserved. This can be seen by expanding our horizons and instead of only looking at Ae^it, we instead replace i by its matrix representation, and then also add the matrix representation for the lie algebra elements of SU(2) i.e. the pauli matrices. Then for QCD use the 8 lie algebra elements of SU(3). This kind of progression seems very natural and intuitive to me, which is why I included this in the original comment. I see math as very distinct from physics. Math is a language for understanding universal truths about abstract concepts, and as far as QM is concerned, I think what it says about nature is that there are a number of symmetries besides the ones found in classical physics. The mathematics of symmetry is all tied up with complex numbers, but it could all be done without it if you really wanted. I believe that by understanding the mathematics of lie groups and lie algebras we gain a much better understanding of the relationship between complex numbers and quantum amplitudes. I even think the question "why are quantum amplitudes complex" potentially raises a number of red herring questions that wont help us understand nature better, when what we really should be focusing on is the role of the mathematics of symmetry in QM, not complex numbers. (P.S. I am going from memory here and may have gotten some details wrong and apologize for being possibly misleading by accident, its hard to communicate this kind of thing but its fun to try)
- auntienomen 8y ago
  You mean O(2), which is isomorphic to U(1). O(1) is a discrete group. You're mistaken that the existence of this U(1) is the reason that charge is conserved. The U(1) you're talking about is the group of units of the complex numbers. It acts trivially in all quantum mechanical calculations, essentially because quantum mechanical states are actually rays in a Hilbert space. There is a different U(1) which is related to charge conservation. It appears in models where one can defined a charge, and acts via $e^{iq}$ on the subspace of states of charge $q$. Note that the action of this group on Hilbert space is non-trivial! It acts differently on different subspaces. In models where this group is a symmetry, by definition, time evolution doesn't alter the charge eigenstates, hence charge is conserved.

calhoun137 8y ago

Great points, let me try and explain better where the group O(1) appears in basic QM. An amplitude in QM is only defined up to a global phase transformation e^it. To be more clear, given some amplitude A and a real number t, we will calculate the same probability for any measurement by using the amplitude Ae^it instead. Well there is the group O(1) right there. What's super interesting is that this fact is literally the reason why charge is conserved. This can be seen by expanding our horizons and instead of only looking at Ae^it, we instead replace i by its matrix representation, and then also add the matrix representation for the lie algebra elements of SU(2) i.e. the pauli matrices. Then for QCD use the 8 lie algebra elements of SU(3). This kind of progression seems very natural and intuitive to me, which is why I included this in the original comment. I see math as very distinct from physics. Math is a language for understanding universal truths about abstract concepts, and as far as QM is concerned, I think what it says about nature is that there are a number of symmetries besides the ones found in classical physics. The mathematics of symmetry is all tied up with complex numbers, but it could all be done without it if you really wanted. I believe that by understanding the mathematics of lie groups and lie algebras we gain a much better understanding of the relationship between complex numbers and quantum amplitudes. I even think the question "why are quantum amplitudes complex" potentially raises a number of red herring questions that wont help us understand nature better, when what we really should be focusing on is the role of the mathematics of symmetry in QM, not complex numbers. (P.S. I am going from memory here and may have gotten some details wrong and apologize for being possibly misleading by accident, its hard to communicate this kind of thing but its fun to try)

auntienomen 8y ago
You mean O(2), which is isomorphic to U(1). O(1) is a discrete group. You're mistaken that the existence of this U(1) is the reason that charge is conserved. The U(1) you're talking about is the group of units of the complex numbers. It acts trivially in all quantum mechanical calculations, essentially because quantum mechanical states are actually rays in a Hilbert space. There is a different U(1) which is related to charge conservation. It appears in models where one can defined a charge, and acts via $e^{iq}$ on the subspace of states of charge $q$. Note that the action of this group on Hilbert space is non-trivial! It acts differently on different subspaces. In models where this group is a symmetry, by definition, time evolution doesn't alter the charge eigenstates, hence charge is conserved.