Q.M., Q.M.+Spin, Q.M.+Relativity; C^1, C^2, C^4, why ?

[Spinnor]

One complex number for quantum mechanics, two complex numbers when we add spin to quantum mechanics, and four complex numbers when we add relativity theory to quantum mechanics.

Can you give me some deeper representation of Nature such that the above is obvious and natural?

Thanks for any help!

U(1) , SU(2) , SU(2) X SU(2)

 

[A. Neumaier]

One complex number for quantum mechanics, two complex numbers when we add spin to quantum mechanics, and four complex numbers when we add relativity theory to quantum mechanics.

Spin distinguishes between up and down --> 2 numbers.
Relativity also between electron and positron --> 2x2 numbers.
Including heavier leptons, we get 3 families --> 2x2x3 numbers.

 

[PhilDSP]

Another possible interpretation:

One complex number represents all possible phase relationships of one object in relation to a given frame of reference (where 2 dimensions are involved).

Two complex numbers represent all possible phase relationships of one object in relation to another object rather than a fixed frame of reference (where 2 dimensions are involved).

Three complex numbers (embedded in a quaternion) represent all possible phase relationships of one object in relation to a given frame of reference (where 3 dimensions are involved). However, those do not uniquely describe the relationship because of the non-commuting properties of rotations in 3 dimensions.

Four complex numbers (embedded in a quaternion) represent all possible phase relationships of one object in relation to a given frame of reference (where 3 dimensions plus time are involved). However, those do not uniquely describe the relationship because of the non-commuting properties of rotations in 3 dimensions.

If multiple complex numbers are used to represent phase in more than 2 dimensions and are not used within a quaternion then they need to obey quaternion algebra (or an equivalent algebra). The Pauli spin matrices are inherently quaternionic.

 

[akhmeteli]

One complex number for quantum mechanics, two complex numbers when we add spin to quantum mechanics, and four complex numbers when we add relativity theory to quantum mechanics.

Can you give me some deeper representation of Nature such that the above is obvious and natural?

I don't know about "natural", but I am not sure any "deeper representation" can make the above "obvious".

First of all, maybe I am splitting hairs, but I would say it's not "One complex number for quantum mechanics", but "One complex function for quantum mechanics", as one complex number cannot describe much, maybe just probability density in one point, but then how is it any better than the probability density itself, which is a real number? A wave function phase in one point does not make much sense.

Second, strictly speaking, one real function may be enough both for the Klein-Gordon equation and for the Dirac equation (with some caveats). See details at https://www.physicsforums.com/showpost.php?p=3008318&postcount=11