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Rotating a spin-2 particle 180° can bring it back to the same quantum state, and a spin-4 particle should be rotated 90° to bring it back to the same quantum state. The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated 180°, and a spin-0 particle can be imagined as sphere, which looks the same ...
As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C 2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).
The spin–statistics theorem implies that all bosons have an integer-valued spin. [2] Scalar bosons are the subset of bosons with zero-valued spin. The name scalar boson arises from quantum field theory, which demands that fields of spin-zero particles transform like a scalar under Lorentz transformation (i.e. are Lorentz invariant).
Hence, Goldstone's original scalar doublet, the massive spin-zero particle, was proposed as the Higgs boson, and is a key building block in the Standard Model. [44] It has no intrinsic spin , and for that reason is classified as a boson with spin-0.
The field can be redefined to give a real scalar field (i.e., a spin-zero particle) θ without any constraint by ϕ = v e i θ {\displaystyle \phi =ve^{i\theta }} where θ is the Nambu–Goldstone boson (actually v θ {\displaystyle v\theta } is) and the U (1) symmetry transformation effects a shift on θ , namely
The (total) spin quantum number has only one value for every elementary particle. Some introductory chemistry textbooks describe m s as the spin quantum number, [6] [7] and s is not mentioned since its value 1 / 2 is a fixed property of the electron; some even use the variable s in place of m s. [5]
In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired. The term 'singlet' originally meant a linked set of particles whose net angular momentum is zero, that is, whose overall spin quantum number =. As a result, there is only one spectral line of a singlet state.
This problem is overcome in different ways depending on particle spin–statistics. For a state of integer spin the negative norm states (known as "unphysical polarization") are set to zero, which makes the use of gauge symmetry necessary. For a state of half-integer spin the argument can be circumvented by having fermionic statistics. [21]