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The reason amplitudes are complex is because a fundamental role is played in quantum theory by symmetry, and the imaginary number in an exponential (e^it) is a

The reason amplitudes are complex is because a fundamental role is played in quantum theory by symmetry, and the imaginary number in an exponential (e^it) is a very algebraically convenient way to capture an important symmetry. In quantum field theory, the lie algebra elements of certain lie groups (U(1), SU(2), and SU(3)) play a fundamental role. For example, the 8 quarks correspond to 8 lie algebra elements of SU(3). A lie group is a continuous symmetry group (such as the rotation of a circle which leave it unchanged, as opposed to a finite group like the rotations of a square) and the lie algebra are the "generators" of lie groups in the same sense that finite groups can be generated, such as the symmetry group of rotations of a square can be generated by a single rotation of 90 degree's. Here is the real kicker, the imaginary number i is actually equivalent in a very strong sense to the lie algebra element of O(1) (which is just the rotations of a circle) and can also be represented by the matrix: 0 -1 1 0 as can be proven by taking the powers of this matrix and the powers of i and simply checking everything.

jessriedel 8y ago
> In quantum field theory, the lie algebra elements of certain lie groups (U(1), SU(2), and SU(3)) play a fundamental role. For example, the 8 quarks correspond to 8 lie algebra elements of SU(3). Those are symmetries of the gauge fields, which apply just as well to classical fields as quantum ones. https://physics.stackexchange.com/questions/352649/does-classical-electrodynamics-have-u1-symmetry-if-yes-how https://physics.stackexchange.com/questions/352649/does-clas... The fields describe possible physical configurations, which is the same for classical mechanics and quantum mechanics. In contrast, the numbers determining the chance of different configurations being realized are complex for quantum mechanics (amplitudes) but real and positive for classical Liouville mechanics (probabilities).
- calhoun137 8y ago
  > the chance of different configurations being realized are complex for quantum mechanics I don't think there is anything essential about complex numbers when it comes to formulating the laws of QM. Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of real numbers at the end of the day. In the wiki article on the Heisenberg picture [1] of QM we read: > By the Stone–von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a basis change in Hilbert space. Then in the article on the Stone-von Neumann theorem [2] we read: > Informally stated, with certain technical assumptions, every representation of the Heisenberg group H2n + 1 is equivalent to the position operators and momentum operators on Rn And in the article on the Heisenberg group [3] we find out that this group consists of 3 by 3 upper triangular matrices with three independent elements that can be taken from any ring. This kind of high level lie group lie algebra approach is something I got really into in the past and continue to be fascinated by. I wish I could explain better why as a result I don't see complex numbers as essential at all in the foundations of quantum theory, but rather, just very convenient for doing calculations. I hope you find the linked wiki articles as interesting as I do. Cheers! [1] https://en.wikipedia.org/wiki/Heisenberg_picture https://en.wikipedia.org/wiki/Heisenberg_picture [2] https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem#Representation_theory_formulation https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theo... [3] https://en.wikipedia.org/wiki/Heisenberg_group https://en.wikipedia.org/wiki/Heisenberg_group
  - jessriedel 8y ago
    > I don't think there is anything essential about complex numbers when it comes to formulating the laws of QM. Everything can be done without complex numbers by using matrices. I'm well aware of this (as this is my profession). My other comment, which you already replied to, agrees that there exist representations of QM that do not feature complex numbers. That's besides the point I was making.
  - chriswarbo 8y ago
    > Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of real numbers at the end of the day. Slightly tangential, but whilst this is a perfectly valid way of thinking about things, to me it seems to be mixing up concepts (e.g. real numbers, complex numbers, matrices, etc.) with encodings-of-concepts-using-other-concepts (in software terms, it's conflating an interface with a particular implementation). In particular, I wouldn't say that a complex number is a pair of real numbers. I'd rather define complex numbers by their algebraic laws (e.g. i × i = -1), and note that some other concepts (like pairs of reals, certain matrices, clifford algebras, etc.) can satisfy these same laws. Similar to how fractions can be encoded as a pair of integers (where "(a, b)" encodes "a ÷ b"), integers can be encoded using a pair of naturals (where "(a, b)" encodes "a - b"), etc. yet we rarely claim that, say, -3 is the pair (0, 3). Likewise a vector/matrix/tensor isn't a 1/2/N dimensional arrangement of numbers. This is a total nit-pick, but notice that you're larger point is basically arguing the same thing w.r.t. quantum mechanics (that it doesn't rely on complex numbers, since other representations exist) ;)
auntienomen 8y ago
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)
AnimalMuppet 8y ago
> For example, the 8 quarks correspond to 8 lie algebra elements of SU(3). Um, what? Last I checked, there are 6 quarks, not 8.

jessriedel 8y ago

> In quantum field theory, the lie algebra elements of certain lie groups (U(1), SU(2), and SU(3)) play a fundamental role. For example, the 8 quarks correspond to 8 lie algebra elements of SU(3). Those are symmetries of the gauge fields, which apply just as well to classical fields as quantum ones. https://physics.stackexchange.com/questions/352649/does-classical-electrodynamics-have-u1-symmetry-if-yes-how https://physics.stackexchange.com/questions/352649/does-clas... The fields describe possible physical configurations, which is the same for classical mechanics and quantum mechanics. In contrast, the numbers determining the chance of different configurations being realized are complex for quantum mechanics (amplitudes) but real and positive for classical Liouville mechanics (probabilities).

calhoun137 8y ago
> the chance of different configurations being realized are complex for quantum mechanics I don't think there is anything essential about complex numbers when it comes to formulating the laws of QM. Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of real numbers at the end of the day. In the wiki article on the Heisenberg picture [1] of QM we read: > By the Stone–von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a basis change in Hilbert space. Then in the article on the Stone-von Neumann theorem [2] we read: > Informally stated, with certain technical assumptions, every representation of the Heisenberg group H2n + 1 is equivalent to the position operators and momentum operators on Rn And in the article on the Heisenberg group [3] we find out that this group consists of 3 by 3 upper triangular matrices with three independent elements that can be taken from any ring. This kind of high level lie group lie algebra approach is something I got really into in the past and continue to be fascinated by. I wish I could explain better why as a result I don't see complex numbers as essential at all in the foundations of quantum theory, but rather, just very convenient for doing calculations. I hope you find the linked wiki articles as interesting as I do. Cheers! [1] https://en.wikipedia.org/wiki/Heisenberg_picture https://en.wikipedia.org/wiki/Heisenberg_picture [2] https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem#Representation_theory_formulation https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theo... [3] https://en.wikipedia.org/wiki/Heisenberg_group https://en.wikipedia.org/wiki/Heisenberg_group
- jessriedel 8y ago
  > I don't think there is anything essential about complex numbers when it comes to formulating the laws of QM. Everything can be done without complex numbers by using matrices. I'm well aware of this (as this is my profession). My other comment, which you already replied to, agrees that there exist representations of QM that do not feature complex numbers. That's besides the point I was making.
- chriswarbo 8y ago
  > Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of real numbers at the end of the day. Slightly tangential, but whilst this is a perfectly valid way of thinking about things, to me it seems to be mixing up concepts (e.g. real numbers, complex numbers, matrices, etc.) with encodings-of-concepts-using-other-concepts (in software terms, it's conflating an interface with a particular implementation). In particular, I wouldn't say that a complex number is a pair of real numbers. I'd rather define complex numbers by their algebraic laws (e.g. i × i = -1), and note that some other concepts (like pairs of reals, certain matrices, clifford algebras, etc.) can satisfy these same laws. Similar to how fractions can be encoded as a pair of integers (where "(a, b)" encodes "a ÷ b"), integers can be encoded using a pair of naturals (where "(a, b)" encodes "a - b"), etc. yet we rarely claim that, say, -3 is the pair (0, 3). Likewise a vector/matrix/tensor isn't a 1/2/N dimensional arrangement of numbers. This is a total nit-pick, but notice that you're larger point is basically arguing the same thing w.r.t. quantum mechanics (that it doesn't rely on complex numbers, since other representations exist) ;)

auntienomen 8y ago

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)

AnimalMuppet 8y ago

> For example, the 8 quarks correspond to 8 lie algebra elements of SU(3). Um, what? Last I checked, there are 6 quarks, not 8.