Fermions and bosons

Those particles with half-integer spins, such as 1/2, 3/2, 5/2, are known as fermions, while those particles with integer spins, such as 0, 1, 2, are known as bosons. The two families of particles obey different rules and broadly have different roles in the world around us. A key distinction between the two families is that fermions obey the Pauli exclusion principle: that is, there cannot be two identical fermions simultaneously having the same quantum numbers (meaning, roughly, having the same position, velocity and spin direction). Fermions obey the rules of Fermi–Dirac statistics. In contrast, bosons obey the rules of Bose–Einstein statistics and have no such restriction, so they may "bunch together" in identical states. Also, composite particles can have spins different from their component particles. For example, a helium-4 atom in the ground state has spin 0 and behaves like a boson, even though the quarks and electrons which make it up are all fermions.

This has some profound consequences:

• Quarks and leptons (including electrons and neutrinos), which make up what is classically known as matter, are all fermions with spin 1/2. The common idea that "matter takes up space" actually comes from the Pauli exclusion principle acting on these particles to prevent the fermions from being in the same quantum state. Further compaction would require electrons to occupy the same energy states, and therefore a kind of pressure (sometimes known as degeneracy pressure of electrons) acts to resist the fermions being overly close.
  Elementary fermions with other spins (3/2, 5/2, etc.) are not known to exist.
• Elementary particles which are thought of as carrying forces are all bosons with spin 1. They include the photon, which carries the electromagnetic force, the gluon (strong force), and the W and Z bosons (weak force). The ability of bosons to occupy the same quantum state is used in the laser, which aligns many photons having the same quantum number (the same direction and frequency), superfluid liquid helium resulting from helium-4 atoms being bosons, and superconductivity, where pairs of electrons (which individually are fermions) act as single composite bosons.
  Elementary bosons with other spins (0, 2, 3, etc.) were not historically known to exist, although they have received considerable theoretical treatment and are well established within their respective mainstream theories. In particular, theoreticians have proposed the graviton (predicted to exist by some quantum gravity theories) with spin 2, and the Higgs boson (explaining electroweak symmetry breaking) with spin 0. Since 2013, the Higgs boson with spin 0 has been considered proven to exist.^[10] It is the first scalar elementary particle (spin 0) known to exist in nature.
• Atomic nuclei have nuclear spin which may be either half-integer or integer, so that the nuclei may be either fermions or bosons.

Spin–statistics theorem

The spin–statistics theorem splits particles into two groups: bosons and fermions, where bosons obey Bose–Einstein statistics, and fermions obey Fermi–Dirac statistics (and therefore the Pauli exclusion principle). Specifically, the theory states that particles with an integer spin are bosons, while all other particles have half-integer spins and are fermions. As an example, electrons have half-integer spin and are fermions that obey the Pauli exclusion principle, while photons have integer spin and do not. The theorem relies on both quantum mechanics and the theory of special relativity, and this connection between spin and statistics has been called "one of the most important applications of the special relativity theory".^[11]

Spin projection quantum number and multiplicity

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum for a spin-s particle measured along any direction can only take on the values^[19]

${\displaystyle S_{i}=\hbar s_{i},\quad s_{i}\in \{-s,-(s-1),\dots ,s-1,s\},}$

where S_i is the spin component along the i-th axis (either x, y, or z), s_i is the spin projection quantum number along the i-th axis, and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z axis:

${\displaystyle S_{z}=\hbar s_{z},\quad s_{z}\in \{-s,-(s-1),\dots ,s-1,s\},}$

where S_z is the spin component along the z axis, s_z is the spin projection quantum number along the z axis.

One can see that there are 2s + 1 possible values of s_z. The number "2s + 1" is the multiplicity of the spin system. For example, there are only two possible values for a spin-1/2 particle: s_z = +1/2 and s_z = −1/2. These correspond to quantum states in which the spin component is pointing in the +z or −z directions respectively, and are often referred to as "spin up" and "spin down". For a spin-3/2 particle, like a delta baryon, the possible values are +3/2, +1/2, −1/2, −3/2.

Vector

A single point in space can rotate continuously without becoming tangled. Notice that after a 360-degree rotation, the spiral flips between clockwise and counterclockwise orientations. It returns to its original configuration after spinning a full 720°.

For a given quantum state, one could think of a spin vector ${\textstyle \langle S\rangle }$ whose components are the expectation values of the spin components along each axis, i.e., ${\textstyle \langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]}$. This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation. It turns out that the spin vector is not very useful in actual quantum-mechanical calculations, because it cannot be measured directly: s_x, s_y and s_z cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a Stern–Gerlach apparatus, the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin-1/2 particles, this probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180°—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%.

As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum-mechanical spin can exhibit phenomena analogous to classical gyroscopic effects. For example, one can exert a kind of "torque" on an electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole moment—see the following section). The result is that the spin vector undergoes precession, just like a classical gyroscope. This phenomenon is known as electron spin resonance (ESR). The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance (NMR) spectroscopy and imaging.

Mathematically, quantum-mechanical spin states are described by vector-like objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin-1/2 particle by 360° does not bring it back to the same quantum state, but to the state with the opposite quantum phase; this is detectable, in principle, with interference experiments. To return the particle to its exact original state, one needs a 720° rotation. (The plate trick and Möbius strip give non-quantum analogies.) A spin-zero particle can only have a single quantum state, even after torque is applied. 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 after whatever angle it is turned through.

Operator

Spin obeys commutation relations^[20] analogous to those of the orbital angular momentum:

$${\displaystyle \left[{\hat {S}}_{j},{\hat {S}}_{k}\right]=i\hbar \varepsilon _{jkl}{\hat {S}}_{l},}$$

where ε_jkl is the Levi-Civita symbol. It follows (as with angular momentum) that the eigenvectors of ${\displaystyle {\hat {S}}^{2}}$ and ${\displaystyle {\hat {S}}_{z}}$ (expressed as kets in the total S basis) are^[2]: 166

$${\displaystyle {\begin{aligned}{\hat {S}}^{2}|s,m_{s}\rangle &=\hbar ^{2}s(s+1)|s,m_{s}\rangle ,\\{\hat {S}}_{z}|s,m_{s}\rangle &=\hbar m_{s}|s,m_{s}\rangle .\end{aligned}}}$$

The spin raising and lowering operators acting on these eigenvectors give

$${\displaystyle {\hat {S}}_{\pm }|s,m_{s}\rangle =\hbar {\sqrt {s(s+1)-m_{s}(m_{s}\pm 1)}}|s,m_{s}\pm 1\rangle ,}$$

where ${\displaystyle {\hat {S}}_{\pm }={\hat {S}}_{x}\pm i{\hat {S}}_{y}}$.^[2]: 166

But unlike orbital angular momentum, the eigenvectors are not spherical harmonics. They are not functions of θ and φ. There is also no reason to exclude half-integer values of s and m_s.

All quantum-mechanical particles possess an intrinsic spin ${\displaystyle s}$ (though this value may be equal to zero). The projection of the spin ${\displaystyle s}$ on any axis is quantized in units of the reduced Planck constant, such that the state function of the particle is, say, not ${\displaystyle \psi =\psi (\mathbf {r} )}$, but ${\displaystyle \psi =\psi (\mathbf {r} ,s_{z})}$, where ${\displaystyle s_{z}}$ can take only the values of the following discrete set:

$${\displaystyle s_{z}\in \{-s\hbar ,-(s-1)\hbar ,\dots ,+(s-1)\hbar ,+s\hbar \}.}$$

One distinguishes bosons (integer spin) and fermions (half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.

Pauli matrices

The quantum-mechanical operators associated with spin-1/2 observables are

$${\displaystyle {\hat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }},}$$

where in Cartesian components

$${\displaystyle S_{x}={\frac {\hbar }{2}}\sigma _{x},\quad S_{y}={\frac {\hbar }{2}}\sigma _{y},\quad S_{z}={\frac {\hbar }{2}}\sigma _{z}.}$$

For the special case of spin-1/2 particles, σ_x, σ_y and σ_z are the three Pauli matrices:

$${\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}$$

Pauli exclusion principle

For systems of N identical particles this is related to the Pauli exclusion principle, which states that its wavefunction ${\displaystyle \psi (\mathbf {r} _{1},\sigma _{1},\dots ,\mathbf {r} _{N},\sigma _{N})}$ must change upon interchanges of any two of the N particles as

$${\displaystyle \psi (\dots ,\mathbf {r} _{i},\sigma _{i},\dots ,\mathbf {r} _{j},\sigma _{j},\dots )=(-1)^{2s}\psi (\dots ,\mathbf {r} _{j},\sigma _{j},\dots ,\mathbf {r} _{i},\sigma _{i},\dots ).}$$

Thus, for bosons the prefactor (−1)^2s will reduce to +1, for fermions to −1. In quantum mechanics all particles are either bosons or fermions. In some speculative relativistic quantum field theories "supersymmetric" particles also exist, where linear combinations of bosonic and fermionic components appear. In two dimensions, the prefactor (−1)^2s can be replaced by any complex number of magnitude 1 such as in the anyon.

The above permutation postulate for N-particle state functions has most important consequences in daily life, e.g. the periodic table of the chemical elements.

Rotations

As described above, quantum mechanics states that components of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum-mechanical description of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis. For instance, for a spin-1/2 particle, we would need two numbers a_±1/2, giving amplitudes of finding it with projection of angular momentum equal to +ħ/2 and −ħ/2, satisfying the requirement

$${\displaystyle |a_{+1/2}|^{2}+|a_{-1/2}|^{2}=1.}$$

For a generic particle with spin s, we would need 2s + 1 such parameters. Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It is clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to phase) to the matrix representing rotation AB. Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices:

$${\displaystyle \sum _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}\left(\sum _{n=-j}^{j}U_{nm}a_{n}\right)^{*}\left(\sum _{k=-j}^{j}U_{km}b_{k}\right),}$$

$${\displaystyle \sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.}$$

Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2).^[21] There is one n-dimensional irreducible representation of SU(2) for each dimension, though this representation is n-dimensional real for odd n and n-dimensional complex for even n (hence of real dimension 2n). For a rotation by angle θ in the plane with normal vector ${\textstyle {\hat {\boldsymbol {\theta }}}}$,

$${\displaystyle U=e^{-{\frac {i}{\hbar }}{\boldsymbol {\theta }}\cdot \mathbf {S} },}$$

where ${\textstyle {\boldsymbol {\theta }}=\theta {\hat {\boldsymbol {\theta }}}}$, and S is the vector of spin operators.

Proof

Working in the coordinate system where ${\textstyle {\hat {\theta }}={\hat {z}}}$, we would like to show that S_x and S_y are rotated into each other by the angle θ. Starting with S_x. Using units where ħ = 1:

$${\displaystyle {\begin{aligned}S_{x}\rightarrow U^{\dagger }S_{x}U&=e^{i\theta S_{z}}S_{x}e^{-i\theta S_{z}}\\&=S_{x}+(i\theta )\left[S_{z},S_{x}\right]+\left({\frac {1}{2!}}\right)(i\theta )^{2}\left[S_{z},\left[S_{z},S_{x}\right]\right]+\left({\frac {1}{3!}}\right)(i\theta )^{3}\left[S_{z},\left[S_{z},\left[S_{z},S_{x}\right]\right]\right]+\cdots \end{aligned}}}$$

Using the spin operator commutation relations, we see that the commutators evaluate to i S_y for the odd terms in the series, and to S_x for all of the even terms. Thus:

$${\displaystyle {\begin{aligned}U^{\dagger }S_{x}U&=S_{x}\left[1-{\frac {\theta ^{2}}{2!}}+\cdots \right]-S_{y}\left[\theta -{\frac {\theta ^{3}}{3!}}\cdots \right]\\&=S_{x}\cos \theta -S_{y}\sin \theta ,\end{aligned}}}$$

as expected. Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension (i.e., for any principal spin quantum number s)^[22]: 164

A generic rotation in 3-dimensional space can be built by compounding operators of this type using Euler angles:

$${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha S_{x}}e^{-i\beta S_{y}}e^{-i\gamma S_{z}}.}$$

An irreducible representation of this group of operators is furnished by the Wigner D-matrix:

$${\displaystyle D_{m'm}^{s}(\alpha ,\beta ,\gamma )\equiv \langle sm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|sm\rangle =e^{-im'\alpha }d_{m'm}^{s}(\beta )e^{-im\gamma },}$$

where

$${\displaystyle d_{m'm}^{s}(\beta )=\langle sm'|e^{-i\beta s_{y}}|sm\rangle }$$

is Wigner's small d-matrix. Note that for γ = 2π and α = β = 0; i.e., a full rotation about the z axis, the Wigner D-matrix elements become

$${\displaystyle D_{m'm}^{s}(0,0,2\pi )=d_{m'm}^{s}(0)e^{-im2\pi }=\delta _{m'm}(-1)^{2m}.}$$

Recalling that a generic spin state can be written as a superposition of states with definite m, we see that if s is an integer, the values of m are all integers, and this matrix corresponds to the identity operator. However, if s is a half-integer, the values of m are also all half-integers, giving (−1)^2m = −1 for all m, and hence upon rotation by 2π the state picks up a minus sign. This fact is a crucial element of the proof of the spin–statistics theorem.

Lorentz transformations

We could try the same approach to determine the behavior of spin under general Lorentz transformations, but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact and therefore does not have any faithful, unitary, finite-dimensional representations.

In case of spin-1/2 particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product that is preserved by this representation. We associate a 4-component Dirac spinor ψ with each particle. These spinors transform under Lorentz transformations according to the law

$${\displaystyle \psi '=\exp {\left({\tfrac {1}{8}}\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi ,}$$

where γ_ν are gamma matrices, and ω_μν is an antisymmetric 4 × 4 matrix parametrizing the transformation. It can be shown that the scalar product

$${\displaystyle \langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi }$$

is preserved. It is not, however, positive-definite, so the representation is not unitary.

Measurement of spin along the x, y, or z axes

Each of the (Hermitian) Pauli matrices of spin-1/2 particles has two eigenvalues, +1 and −1. The corresponding normalized eigenvectors are

$${\displaystyle {\begin{array}{lclc}\psi _{x+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{1}\end{pmatrix}},&\psi _{x-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-1}\end{pmatrix}},\\\psi _{y+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{i}\end{pmatrix}},&\psi _{y-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-i}\end{pmatrix}},\\\psi _{z+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{z}=&{\begin{pmatrix}1\\0\end{pmatrix}},&\psi _{z-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{z}=&{\begin{pmatrix}0\\1\end{pmatrix}}.\end{array}}}$$

(Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign. In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as SymPy; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.)

By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the x, y, or z axis can only yield an eigenvalue of the corresponding spin operator (S_x, S_y or S_z) on that axis, i.e. ħ/2 or –ħ/2. The quantum state of a particle (with respect to spin), can be represented by a two-component spinor:

$${\displaystyle \psi ={\begin{pmatrix}a+bi\\c+di\end{pmatrix}}.}$$

When the spin of this particle is measured with respect to a given axis (in this example, the x axis), the probability that its spin will be measured as ħ/2 is just ${\displaystyle {\big |}\langle \psi _{x+}|\psi \rangle {\big |}^{2}}$. Correspondingly, the probability that its spin will be measured as –ħ/2 is just ${\displaystyle {\big |}\langle \psi _{x-}|\psi \rangle {\big |}^{2}}$. Following the measurement, the spin state of the particle collapses into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to have a given eigenvalue, all measurements will yield the same eigenvalue (since ${\displaystyle {\big |}\langle \psi _{x+}|\psi _{x+}\rangle {\big |}^{2}=1}$, etc.), provided that no measurements of the spin are made along other axes.

Measurement of spin along an arbitrary axis

The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let u = (u_x, u_y, u_z) be an arbitrary unit vector. Then the operator for spin in this direction is simply

$${\displaystyle S_{u}={\frac {\hbar }{2}}(u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z}).}$$

The operator S_u has eigenvalues of ±ħ/2, just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three x-, y-, z-axis directions.

A normalized spinor for spin-1/2 in the (u_x, u_y, u_z) direction (which works for all spin states except spin down, where it will give 0/0) is

$${\displaystyle {\frac {1}{\sqrt {2+2u_{z}}}}{\begin{pmatrix}1+u_{z}\\u_{x}+iu_{y}\end{pmatrix}}.}$$

The above spinor is obtained in the usual way by diagonalizing the σ_u matrix and finding the eigenstates corresponding to the eigenvalues. In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity.

Compatibility of spin measurements

Since the Pauli matrices do not commute, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the x axis, and we then measure the spin along the y axis, we have invalidated our previous knowledge of the x axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that

$${\displaystyle {\big |}\langle \psi _{x\pm }|\psi _{y\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{x\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{y\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\tfrac {1}{2}}.}$$

So when physicists measure the spin of a particle along the x axis as, for example, ħ/2, the particle's spin state collapses into the eigenstate ${\displaystyle |\psi _{x+}\rangle }$. When we then subsequently measure the particle's spin along the y axis, the spin state will now collapse into either ${\displaystyle |\psi _{y+}\rangle }$ or ${\displaystyle |\psi _{y-}\rangle }$, each with probability 1/2. Let us say, in our example, that we measure −ħ/2. When we now return to measure the particle's spin along the x axis again, the probabilities that we will measure ħ/2 or −ħ/2 are each 1/2 (i.e. they are ${\displaystyle {\big |}\langle \psi _{x+}|\psi _{y-}\rangle {\big |}^{2}}$ and ${\displaystyle {\big |}\langle \psi _{x-}|\psi _{y-}\rangle {\big |}^{2}}$ respectively). This implies that the original measurement of the spin along the x axis is no longer valid, since the spin along the x axis will now be measured to have either eigenvalue with equal probability.

Higher spins

The spin-1/2 operator S = ħ/2σ forms the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher-spin systems in three spatial dimensions can be calculated for arbitrarily large s using this spin operator and ladder operators. For example, taking the Kronecker product of two spin-1/2 yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 (triplet states) and a 1-dimensional spin-0 representation (singlet state).

The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis:

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G_n is defined to consist of all n-fold tensor products of Pauli matrices.

The analog formula of Euler's formula in terms of the Pauli matrices

$${\displaystyle {\hat {R}}(\theta ,{\hat {\mathbf {n} }})=e^{i{\frac {\theta }{2}}{\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}}=I\cos {\frac {\theta }{2}}+i\left({\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}\right)\sin {\frac {\theta }{2}}}$$

for higher spins is tractable, but less simple.^[23]