Spin (physics)

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Spin is an intrinsic form of angular momentum carried by elementary particles, and thus by composite particles such as hadrons, atomic nuclei, and atoms. Spin should not be understood as in the "rotating internal mass" sense: spin is a quantized wave property.

The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The existence of the electron spin can also be inferred theoretically from the spin–statistics theorem and from the Pauli exclusion principle—and vice versa, given the particular spin of the electron, one may derive the Pauli exclusion principle.

Spin is described mathematically as a vector for some particles such as photons, and as spinors and bispinors for other particles such as electrons. Spinors and bispinors behave similarly to vectors: they have definite magnitudes and change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change. These are indicated by assigning the particle a spin quantum number.

The SI unit of spin is the same as classical angular momentum (i.e., N·m·s, J·s, or kg·m²·s⁻¹). In practice, spin is usually given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ, which has the same dimensions as angular momentum. Often, the "spin quantum number" is simply called "spin".

Relation to classical rotation

The very earliest models for electron spin imagined a rotating charged mass, but this model fails when examined in detail: the required space distribution does not match limits on the electron radius: the required rotation speed exceeds the speed of light. In the Standard Model, the fundamental particles are all considered "point-like": they have their effects through the field that surrounds them. Any model for spin based on mass rotation would need to be consistent with that model.

The classical analog for quantum spin is a circulation of energy or momentum-density in the particle wave field: "spin is essentially a wave property". This same concept of spin can be applied to gravity waves in water: "spin is generated by subwavelength circular motion of water particles".

Photon spin is the quantum-mechanical description of light polarization, where spin +1 and spin −1 represent two opposite directions of circular polarization. Thus, light of a defined circular polarization consists of photons with the same spin, either all +1 or all −1. Spin represents polarization for other vector bosons as well.

Relation to orbital angular momentum

As the name suggests, spin was originally conceived as the rotation of a particle around some axis. Historically orbital angular momentum related to particle orbits. While the names based on mechanical models have survived, the physical explanation has not. Quantization fundamentally alters the character of both spin and orbital angular momentum.

Since elementary particles are point-like, self-rotation is not well-defined for them. However, spin implies that the phase of the particle depends on the angle as ${\displaystyle e^{iS\theta }}$, for rotation of angle θ around the axis parallel to the spin S. This is equivalent to the quantum-mechanical interpretation of momentum as phase dependence in the position, and of orbital angular momentum as phase dependence in the angular position.

For fermions, the picture is less clear. Angular velocity is equal by Ehrenfest theorem to the derivative of the Hamiltonian to its conjugate momentum, which is the total angular momentum operator J = L + S. Therefore, if the Hamiltonian H is dependent upon the spin S, dH/dS is non-zero, and the spin causes angular velocity, and hence actual rotation, i.e. a change in the phase-angle relation over time. However, whether this holds for free electron is ambiguous, since for an electron, S² is constant, and therefore it is a matter of interpretation whether the Hamiltonian includes such a term. Nevertheless, spin appears in the Dirac equation, and thus the relativistic Hamiltonian of the electron, treated as a Dirac field, can be interpreted as including a dependence in the spin S. Under this interpretation, free electrons also self-rotate, with the zitterbewegung effect understood as this rotation.

Quantum number

Main article: Spin quantum number

Spin obeys the mathematical laws of angular momentum quantization. The specific properties of spin angular momenta include:

• Spin quantum numbers may take half-integer values.
• Although the direction of its spin can be changed, the magnitude of the spin of an elementary particle cannot be changed.
• The spin of a charged particle is associated with a magnetic dipole moment with a g-factor that differs from 1. (In the classical context, this would imply the internal charge and mass distributions differing for a rotating object.)

The conventional definition of the spin quantum number is s = n/2, where n can be any non-negative integer. Hence the allowed values of s are 0, 1/2, 1, 3/2, 2, etc. The value of s for an elementary particle depends only on the type of particle and cannot be altered in any known way (in contrast to the spin direction described below). The spin angular momentum S of any physical system is quantized. The allowed values of S are

$${\displaystyle S=\hslash \sqrt{s(s+1)}=\frac{h}{2\pi }\sqrt{\frac{n}{2}\frac{(n+2)}{2}}=\frac{h}{4\pi }\sqrt{n(n+2)},}$$

where h is the Planck constant, and $\hslash =\frac{h}{2\pi }$ is the reduced Planck constant. In contrast, orbital angular momentum can only take on integer values of s; i.e., even-numbered values of n.

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. 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

Main article: 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".

Magnetic moments

Main article: Spin magnetic moment

Schematic diagram depicting the spin of the neutron as the black arrow and magnetic field lines associated with the neutron magnetic moment. The neutron has a negative magnetic moment. While the spin of the neutron is upward in this diagram, the magnetic field lines at the center of the dipole are downward.

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern–Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves.

The intrinsic magnetic moment μ of a spin-1/2 particle with charge q, mass m, and spin angular momentum S, is

${\displaystyle \mathit{\mu }=\frac{g_{\text{s}}q}{2m}\mathbf{S},}$

where the dimensionless quantity g_s is called the spin g-factor. For exclusively orbital rotations it would be 1 (assuming that the mass and the charge occupy spheres of equal radius).

The electron, being a charged elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g-factor, which has been experimentally determined to have the value −2.00231930436256(35), with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation. The value of 2 arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties, and the deviation from −2 arises from the electron's interaction with the surrounding electromagnetic field, including its own field.

Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions.

Neutrinos are both elementary and electrically neutral. The minimally extended Standard Model that takes into account non-zero neutrino masses predicts neutrino magnetic moments of:

${\displaystyle {\mu }_{\nu }\approx 3\times {10}^{-19}{\mu }_{\text{B}}\frac{m_{\nu }{c}^2}{\text{eV}},}$

where the μ_ν are the neutrino magnetic moments, m_ν are the neutrino masses, and μ_B is the Bohr magneton. New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments. It can be shown in a model-independent way that neutrino magnetic moments larger than about 10⁻¹⁴ μ_B are "unnatural" because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses are known to be at most about 1 eV/c², fine-tuning would be necessary in order to prevent large contributions to the neutrino mass via radiative corrections. The measurement of neutrino magnetic moments is an active area of research. Experimental results have put the neutrino magnetic moment at less than 1.2×10⁻¹⁰ times the electron's magnetic moment.

On the other hand elementary particles with spin but without electric charge, such as a photon or a Z boson, do not have a magnetic moment.

Curie temperature and loss of alignment

In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction, with the overall average being very near zero. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments spontaneously align locally, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.

In paramagnetic materials, the magnetic dipole moments of individual atoms will partially align with an externally applied magnetic field. In diamagnetic materials, on the other hand, the magnetic dipole moments of individual atoms align oppositely to any externally applied magnetic field, even if it requires energy to do so.

The study of the behavior of such "spin models" is a thriving area of research in condensed matter physics. For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to interesting results in the theory of phase transitions.

Direction

Further information: Angular momentum operator

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

${\displaystyle S_i=\hslash s_i,s_i\in \{-s,-(s-1),\dots ,s-1,s\},}$

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:

${\displaystyle S_z=\hslash s_z,s_z\in \{-s,-(s-1),\dots ,s-1,s\},}$

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 $⟨S⟩$ whose components are the expectation values of the spin components along each axis, i.e., $⟨S⟩=[⟨S_x⟩,⟨S_y⟩,⟨S_z⟩]$. 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.

Mathematical formulation

Operator

Spin obeys commutation relations analogous to those of the orbital angular momentum:

$${\displaystyle \left[{\hat{S}}_j,{\hat{S}}_k\right]=i\hslash {\epsilon }_{jkl}{\hat{S}}_l,}$$

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

$${\displaystyle \begin{array}{rl}{\hat{S}}^2|s,m_s⟩& ={\hslash }^{2}s(s+1)|s,m_s⟩,\\ {\hat{S}}_z|s,m_s⟩& =\hslash m_s|s,m_s⟩.\end{array}}$$

The spin raising and lowering operators acting on these eigenvectors give

$${\displaystyle {\hat{S}}_{\pm }|s,m_s⟩=\hslash \sqrt{s(s+1)-m_s(m_s\pm 1)}|s,m_s\pm 1⟩,}$$

where ${\displaystyle {\hat{S}}_{\pm }={\hat{S}}_x\pm i{\hat{S}}_y}$.

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:

$${\displaystyle s_z\in \{-s\hslash ,-(s-1)\hslash ,\dots ,+(s-1)\hslash ,+s\hslash \}.}$$

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

Main article: Pauli matrices

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

$${\displaystyle \hat{\mathbf{S}}=\frac{\hslash }{2}\mathit{\sigma },}$$

where in Cartesian components

$${\displaystyle S_x=\frac{\hslash }{2}{\sigma }_x,S_y=\frac{\hslash }{2}{\sigma }_y,S_z=\frac{\hslash }{2}{\sigma }_z.}$$

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

$${\displaystyle {\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).}$$

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

See also: Symmetry in quantum mechanics

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:

$${\displaystyle \sum _{m=-j}^j{a}_m^{\ast }{b}_m=\sum _{m=-j}^j{\left(\sum _{n=-j}^j{U}_{nm}{a}_n\right)}^{\ast }\left(\sum _{k=-j}^j{U}_{km}{b}_k\right),}$$

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

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). 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 $\hat{\mathit{\theta }}$,

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

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

Proof

Working in the coordinate system where $\hat{\theta }=\hat{z}$, 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:

$${\displaystyle \begin{array}{rl}{S}_x\to U^{†}{S}_{x}U& =e^{i\theta S_z}{S}_x{e}^{-i\theta S_z}\\ & =S_x+(i\theta )\left[S_z,S_x\right]+\left(\frac{1}{2!}\right)(i\theta {)}^2\left[S_z,\left[S_z,S_x\right]\right]+\left(\frac{1}{3!}\right)(i\theta {)}^3\left[S_z,\left[S_z,\left[S_z,S_x\right]\right]\right]+\cdots \end{array}}$$

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:

$${\displaystyle \begin{array}{rl}{U}^{†}{S}_{x}U& =S_x\left[1-\frac{{\theta }^2}{2!}+\cdots \right]-S_y\left[\theta -\frac{{\theta }^3}{3!}\cdots \right]\\ & =S_x\cos \theta -S_y\sin \theta ,\end{array}}$$

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).

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:

$${\displaystyle D_{m^{\prime }m}^s(\alpha ,\beta ,\gamma )\equiv ⟨s{m}^{\prime }|\mathcal{R}(\alpha ,\beta ,\gamma )|sm⟩=e^{-i{m}^{\prime }\alpha }{d}_{m^{\prime }m}^s(\beta )e^{-im\gamma },}$$

where

$${\displaystyle d_{m^{\prime }m}^s(\beta )=⟨s{m}^{\prime }|e^{-i\beta s_y}|sm⟩}$$

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

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

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

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

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

$${\displaystyle ⟨\psi |\varphi ⟩=\overline{\psi }\varphi ={\psi }^{†}{\gamma }_0\varphi }$$

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

$${\displaystyle \begin{array}{lclc}{\psi }_{x+}={|\frac{1}{2},\frac{+1}{2}⟩}_x={\displaystyle \frac{1}{\sqrt{2}}}& \left(\begin{array}{c}1\\ 1\end{array}\right),& {\psi }_{x-}={|\frac{1}{2},\frac{-1}{2}⟩}_x={\displaystyle \frac{1}{\sqrt{2}}}& \left(\begin{array}{c}1\\ -1\end{array}\right),\\ {\psi }_{y+}={|\frac{1}{2},\frac{+1}{2}⟩}_y={\displaystyle \frac{1}{\sqrt{2}}}& \left(\begin{array}{c}1\\ i\end{array}\right),& {\psi }_{y-}={|\frac{1}{2},\frac{-1}{2}⟩}_y={\displaystyle \frac{1}{\sqrt{2}}}& \left(\begin{array}{c}1\\ -i\end{array}\right),\\ {\psi }_{z+}={|\frac{1}{2},\frac{+1}{2}⟩}_z=& \left(\begin{array}{c}1\\ 0\end{array}\right),& {\psi }_{z-}={|\frac{1}{2},\frac{-1}{2}⟩}_z=& \left(\begin{array}{c}0\\ 1\end{array}\right).\end{array}}$$

(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:

$${\displaystyle \psi =\left(\begin{array}{c}a+bi\\ c+di\end{array}\right).}$$

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 ${\displaystyle |⟨{\psi }_{x+}|\psi ⟩{|}^2}$. Correspondingly, the probability that its spin will be measured as –ħ/2 is just ${\displaystyle |⟨{\psi }_{x-}|\psi ⟩{|}^2}$. 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 ${\displaystyle |⟨{\psi }_{x+}|{\psi }_{x+}⟩{|}^2=1}$, 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

$${\displaystyle S_u=\frac{\hslash }{2}(u_x{\sigma }_x+u_y{\sigma }_y+u_z{\sigma }_z).}$$

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

$${\displaystyle \frac{1}{\sqrt{2+2u_z}}\left(\begin{array}{c}1+u_z\\ u_x+i{u}_y\end{array}\right).}$$

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

$${\displaystyle |⟨{\psi }_{x\pm }|{\psi }_{y\pm }⟩{|}^2=|⟨{\psi }_{x\pm }|{\psi }_{z\pm }⟩{|}^2=|⟨{\psi }_{y\pm }|{\psi }_{z\pm }⟩{|}^2=\frac{1}{2}.}$$

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 ${\displaystyle |{\psi }_{x+}⟩}$. When we then subsequently measure the particle's spin along the y axis, the spin state will now collapse into either ${\displaystyle |{\psi }_{y+}⟩}$ or ${\displaystyle |{\psi }_{y-}⟩}$, 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 ${\displaystyle |⟨{\psi }_{x+}|{\psi }_{y-}⟩{|}^2}$ and ${\displaystyle |⟨{\psi }_{x-}|{\psi }_{y-}⟩{|}^2}$ 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

See also: 3D rotation group § A note on Lie algebras

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:

1. For spin 1 they are
   $${\displaystyle \begin{array}{rlrlrlrl}{S}_x& =\frac{\hslash }{\sqrt{2}}\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right),& {|1,+1⟩}_x& =\frac{1}{2}\left(\begin{array}{c}1\\ \sqrt{2}\\ 1\end{array}\right),& {|1,0⟩}_x& =\frac{1}{\sqrt{2}}\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right),& {|1,-1⟩}_x& =\frac{1}{2}\left(\begin{array}{c}1\\ -\sqrt{2}\\ 1\end{array}\right)\\ S_y& =\frac{\hslash }{\sqrt{2}}\left(\begin{array}{ccc}0& -i& 0\\ i& 0& -i\\ 0& i& 0\end{array}\right),& {|1,+1⟩}_y& =\frac{1}{2}\left(\begin{array}{c}-1\\ -i\sqrt{2}\\ 1\end{array}\right),& {|1,0⟩}_y& =\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right),& {|1,-1⟩}_y& =\frac{1}{2}\left(\begin{array}{c}-1\\ i\sqrt{2}\\ 1\end{array}\right)\\ S_z& =\hslash \left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& -1\end{array}\right),& {|1,+1⟩}_z& =\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),& {|1,0⟩}_z& =\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),& {|1,-1⟩}_z& =\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\end{array}}$$

   
2. For spin 3/2 they are
   $${\displaystyle \begin{array}{lclcccccc}{S}_x=\frac{\hslash }{2}\left(\begin{array}{cccc}0& \sqrt{3}& 0& 0\\ \sqrt{3}& 0& 2& 0\\ 0& 2& 0& \sqrt{3}\\ 0& 0& \sqrt{3}& 0\end{array}\right),& {|\frac{3}{2},\frac{+3}{2}⟩}_x=& \frac{1}{2\sqrt{2}}\left(\begin{array}{c}1\\ \sqrt{3}\\ \sqrt{3}\\ 1\end{array}\right),& {|\frac{3}{2},\frac{+1}{2}⟩}_x=& \frac{1}{2\sqrt{2}}\left(\begin{array}{c}-\sqrt{3}\\ -1\\ 1\\ \sqrt{3}\end{array}\right),& {|\frac{3}{2},\frac{-1}{2}⟩}_x=& \frac{1}{2\sqrt{2}}\left(\begin{array}{c}\sqrt{3}\\ -1\\ -1\\ \sqrt{3}\end{array}\right),& {|\frac{3}{2},\frac{-3}{2}⟩}_x=& \frac{1}{2\sqrt{2}}\left(\begin{array}{c}-1\\ \sqrt{3}\\ -\sqrt{3}\\ 1\end{array}\right)\\ S_y=\frac{\hslash }{2}\left(\begin{array}{cccc}0& -i\sqrt{3}& 0& 0\\ i\sqrt{3}& 0& -2i& 0\\ 0& 2i& 0& -i\sqrt{3}\\ 0& 0& i\sqrt{3}& 0\end{array}\right),& {|\frac{3}{2},\frac{+3}{2}⟩}_y=& \frac{1}{2\sqrt{2}}\left(\begin{array}{c}i\\ -\sqrt{3}\\ -i\sqrt{3}\\ 1\end{array}\right),& {|\frac{3}{2},\frac{+1}{2}⟩}_y=& \frac{1}{2\sqrt{2}}\left(\begin{array}{c}-i\sqrt{3}\\ 1\\ -i\\ \sqrt{3}\end{array}\right),& {|\frac{3}{2},\frac{-1}{2}⟩}_y=& \frac{1}{2\sqrt{2}}\left(\begin{array}{c}i\sqrt{3}\\ 1\\ i\\ \sqrt{3}\end{array}\right),& {|\frac{3}{2},\frac{-3}{2}⟩}_y=& \frac{1}{2\sqrt{2}}\left(\begin{array}{c}-i\\ -\sqrt{3}\\ i\sqrt{3}\\ 1\end{array}\right)\\ S_z=\frac{\hslash }{2}\left(\begin{array}{cccc}3& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -3\end{array}\right),& {|\frac{3}{2},\frac{+3}{2}⟩}_z=& \left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right),& {|\frac{3}{2},\frac{+1}{2}⟩}_z=& \left(\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right),& {|\frac{3}{2},\frac{-1}{2}⟩}_z=& \left(\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right),& {|\frac{3}{2},\frac{-3}{2}⟩}_z=& \left(\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right)\end{array}}$$

   
3. For spin 5/2 they are
   $${\displaystyle \begin{array}{rl}{\mathit{S}}_x& =\frac{\hslash }{2}\left(\begin{array}{cccccc}0& \sqrt{5}& 0& 0& 0& 0\\ \sqrt{5}& 0& 2\sqrt{2}& 0& 0& 0\\ 0& 2\sqrt{2}& 0& 3& 0& 0\\ 0& 0& 3& 0& 2\sqrt{2}& 0\\ 0& 0& 0& 2\sqrt{2}& 0& \sqrt{5}\\ 0& 0& 0& 0& \sqrt{5}& 0\end{array}\right),\\ {\mathit{S}}_y& =\frac{\hslash }{2}\left(\begin{array}{cccccc}0& -i\sqrt{5}& 0& 0& 0& 0\\ i\sqrt{5}& 0& -2i\sqrt{2}& 0& 0& 0\\ 0& 2i\sqrt{2}& 0& -3i& 0& 0\\ 0& 0& 3i& 0& -2i\sqrt{2}& 0\\ 0& 0& 0& 2i\sqrt{2}& 0& -i\sqrt{5}\\ 0& 0& 0& 0& i\sqrt{5}& 0\end{array}\right),\\ {\mathit{S}}_z& =\frac{\hslash }{2}\left(\begin{array}{cccccc}5& 0& 0& 0& 0& 0\\ 0& 3& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& -3& 0\\ 0& 0& 0& 0& 0& -5\end{array}\right).\end{array}}$$

   
4. The generalization of these matrices for arbitrary spin s is
   $${\displaystyle \begin{array}{rl}{\left(S_x\right)}_{ab}& =\frac{\hslash }{2}\left({\delta }_{a,b+1}+{\delta }_{a+1,b}\right)\sqrt{(s+1)(a+b-1)-ab},\\ {\left(S_y\right)}_{ab}& =\frac{i\hslash }{2}\left({\delta }_{a,b+1}-{\delta }_{a+1,b}\right)\sqrt{(s+1)(a+b-1)-ab},\\ {\left(S_z\right)}_{ab}& =\hslash (s+1-a){\delta }_{a,b}=\hslash (s+1-b){\delta }_{a,b},\end{array}}$$

   
   where indices ${\displaystyle a,b}$ are integer numbers such that
   $${\displaystyle 1\le a\le 2s+1,1\le b\le 2s+1.}$$

   

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

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

for higher spins is tractable, but less simple.

Parity

In tables of the spin quantum number s for nuclei or particles, the spin is often followed by a "+" or "−". This refers to the parity with "+" for even parity (wave function unchanged by spatial inversion) and "−" for odd parity (wave function negated by spatial inversion). For example, see the isotopes of bismuth, in which the list of isotopes includes the column nuclear spin and parity. For Bi-209, the longest-lived isotope, the entry 9/2– means that the nuclear spin is 9/2 and the parity is odd.

Applications

Spin has important theoretical implications and practical applications. Well-established direct applications of spin include:

• Nuclear magnetic resonance (NMR) spectroscopy in chemistry;
• Electron spin resonance (ESR or EPR) spectroscopy in chemistry and physics;
• Magnetic resonance imaging (MRI) in medicine, a type of applied NMR, which relies on proton spin density;
• Giant magnetoresistive (GMR) drive-head technology in modern hard disks.

Electron spin plays an important role in magnetism, with applications for instance in computer memories. The manipulation of nuclear spin by radio-frequency waves (nuclear magnetic resonance) is important in chemical spectroscopy and medical imaging.

Spin–orbit coupling leads to the fine structure of atomic spectra, which is used in atomic clocks and in the modern definition of the second. Precise measurements of the g-factor of the electron have played an important role in the development and verification of quantum electrodynamics. Photon spin is associated with the polarization of light (photon polarization).

An emerging application of spin is as a binary information carrier in spin transistors. The original concept, proposed in 1990, is known as Datta–Das spin transistor. Electronics based on spin transistors are referred to as spintronics. The manipulation of spin in dilute magnetic semiconductor materials, such as metal-doped ZnO or TiO₂ imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.

There are many indirect applications and manifestations of spin and the associated Pauli exclusion principle, starting with the periodic table of chemistry.

History

Wolfgang Pauli lecturing

Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924, Wolfgang Pauli introduced what he called a "two-valuedness not describable classically" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can have the same quantum state in the same quantum system.

The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea.

In the autumn of 1925, the same thought came to Dutch physicists George Uhlenbeck and Samuel Goudsmit at Leiden University. Under the advice of Paul Ehrenfest, they published their results. It met a favorable response, especially after Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished results). This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.

Mathematically speaking, a fiber bundle description is needed. The tangent bundle effect is additive and relativistic; that is, it vanishes if c goes to infinity. It is one half of the value obtained without regard for the tangent-space orientation, but with opposite sign. Thus the combined effect differs from the latter by a factor two (Thomas precession, known to Ludwik Silberstein in 1914).

Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics invented by Schrödinger and Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators and introduced a two-component spinor wave-function. Uhlenbeck and Goudsmit treated spin as arising from classical rotation, while Pauli emphasized, that spin is a non-classical and intrinsic property.

Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac published the Dirac equation, which described the relativistic electron. In the Dirac equation, a four-component spinor (known as a "Dirac spinor") was used for the electron wave-function. Relativistic spin explained gyromagnetic anomaly, which was (in retrospect) first observed by Samuel Jackson Barnett in 1914 (see Einstein–de Haas effect). In 1940, Pauli proved the spin–statistics theorem, which states that fermions have half-integer spin, and bosons have integer spin.

In retrospect, the first direct experimental evidence of the electron spin was the Stern–Gerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1927.

Transgender health care

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Transgender_health_care

Transgender health care includes the prevention, diagnosis and treatment of physical and mental health conditions, as well as sex reassignment therapies, for transgender individuals. A major component of transgender health care is gender-affirming care, the medical aspect of gender transition. Questions implicated in transgender health care include gender variance, sex reassignment therapy, health risks (in relation to violence and mental health), and access to healthcare for trans people in different countries around the world.

Gender variance and medicine

Gender variance is defined in medical literature as "gender identity, expression, or behavior that falls outside of culturally defined norms associated with a specific gender". For centuries, gender variance was seen by medicine as a pathology. The World Health Organization identified gender dysphoria as a mental disorder in the International Classification of Diseases (ICD) until 2018. Gender dysphoria was also listed in the Diagnostic and Statistical Manual of Mental Disorders (DSM-5) of the American Psychiatric Association, where it was previously called "transsexualism" and "gender identity disorder".

In 2018, the ICD-11 included the term "gender incongruence" as "marked and persistent incongruence between an individual's experienced gender and the assigned sex", where gender variant behaviour and preferences do not necessarily imply a medical diagnosis. However, the difference between "gender dysphoria" and "gender incongruence" is not always clear in the medical literature.

Some studies posit that treating gender variance as a medical condition has negative effects on the health of transgender people and claim that assumptions of coexisting psychiatric symptoms should be avoided. Other studies argue that gender incongruence diagnosis may be important and even positive for transgender people at the individual and social level.

As there are various ways of classifying or characterizing those who are either diagnosed or self-affirm as transgender individuals, the literature cannot clearly estimate how prevalent these experiences are within the total population. The results of a recent systematic review highlight the need to standardize the scope and methodology related to data collection of those presenting as transgender.

Gender-affirming care

Various options are available for transgender people to pursue physical transition. There have been options for transitioning for transgender individuals since 1917. Gender-affirming care helps people to change their physical appearance and/or sex characteristics to accord with their gender identity; it includes hormone replacement therapy and gender-affirming surgery. While many transgender people do elect to transition physically, every transgender person has different needs and, as such, there is no required transition plan. Preventive health care is a crucial part of transitioning and a primary care physician is recommended for transgender people who are transitioning.

Eligibility

In the 11th version of the International Classification of Diseases (ICD-11), the diagnosis is known as gender incongruence. ICD-11 states that "Gender variant behaviour and preferences alone are not a basis for assigning the diagnosis."

The US Diagnostic and Statistical Manual of Mental Disorders (DSM) names it gender dysphoria (in version 5). Some people who are validly diagnosed have no desire for all or some parts of sex reassignment therapy, particularly genital reassignment surgery, and/or are not appropriate candidates for such treatment.

The general standard for diagnosing, as well as treating, gender dysphoria is outlined in the WPATH Standards of Care for the Health of Transsexual, Transgender, and Gender Nonconforming People. As of February 2023, the most recent version of the standards is Version 8. According to the standards of care, "Gender Dysphoria describes a state of distress or discomfort that may be experienced because a person's gender identity differs from that which is physically and/or socially attributed to their sex assigned at birth… Not all transgender and gender diverse people experience gender dysphoria." Gender nonconformity is not the same as gender dysphoria; nonconformity, according to the standards of care, is not a pathology and does not require medical treatment.

The informed consent model is an alternative to the standard WPATH approach which does not require a person seeking transition related medical treatment to undergo formal assessment of their mental health or gender dysphoria. Arguments in favor of this model describe required assessments as gatekeeping, dehumanizing, pathologizing, and reinforcing a reductive perception of transgender experiences. Informed consent approaches include conversations between the medical provider and person seeking care on the details of risks and outcomes, current understandings of scientific research, and how the provider can best assist the person in making decisions.

Local standards of care exist in many countries.

Eligibility for different stages of treatment

While a mental health assessment is required by the standards of care, psychotherapy is not an absolute requirement but is highly recommended.

Hormone replacement therapy is to be initiated from a qualified health professional. The general requirements, according to the WPATH standards, include:

1. Persistent, well-documented gender dysphoria;
2. Capacity to make a fully informed decision and to consent for treatment;
3. Age of majority in a given country (however, the WPATH standards of care provide separate discussion of children and adolescents);
4. If significant medical or mental health concerns are present, they must be reasonably well-controlled.

Often, at least a certain period of psychological counseling is required before initiating hormone replacement therapy, as is a period of living in the desired gender role, if possible, to ensure that they can psychologically function in that life-role. On the other hand, some clinics provide hormone therapy based on informed consent alone.

Eligibility of minors

While the WPATH standards of care generally require the patient to have reached the age of majority, they include a separate section devoted to children and adolescents.

Hormone replacement therapy

Main article: Transgender hormone therapy

A transgender woman before and after two years of hormone replacement therapy.

Hormone replacement therapy (HRT) is primarily concerned with alleviating gender dysphoria in transgender people. Trans women are usually treated with estrogen and complementary anti-androgenic therapy. This therapy induces breast formation, reduces male hair pattern growth, and changes fat distribution, also leading to a decreased testicular size and erectile function. Trans men are normally treated with exogenous testosterone, which is expected to cease menses, to increase facial and body hair, to cause changes in skin and in fat distribution, and to increase muscle mass and libido. After at least three months, other effects are expected, such as the deepening of the voice and changes in sexual organs (such as atrophy of vaginal tissues, and increased clitoral size). Regular monitoring by an endocrinologist is a strong recommendation to ensure the safety of individuals as they transition.

Access to hormone replacement therapy has been shown to improve quality of life for people in the female-to-male community when compared to female-to-male people who do not have access to hormone replacement therapy. Despite the improvement in quality of life, there are still dangers with hormone replacement therapy, in particular with self-medication. An examination of the use of self-medication found that people who self-medicated were more likely to experience adverse health effects from preexisting conditions such as high blood pressure as well as slower development of desired secondary sex characteristics.

Hormone therapy for transgender individuals has been shown in medical literature to be safe, when supervised by a qualified medical professional.

Transgender people seeking surgery may be informed they will need to take hormones for the rest of their life if they want to maintain the feminizing effects of oestrogen or the masculinizing effects of testosterone. Their dose of hormones will usually be reduced, but it should still be enough to produce the effects that they need and to keep them well, and to protect them against osteoporosis (thinning of the bones) as they get older. If they are still on hormone blockers, they will stop taking them altogether.

Monitoring of risk factors associated with hormone replacement therapy, such as prolactin levels in transgender women and polycythemia levels in transgender men, are crucial for the preventive health care of transgender people taking these treatments.

On July 1, 2022, the FDA issued an update that gonadotropin-releasing hormone agonists, drugs that are approved for treating precocious puberty, may be a risk factor for developing pseudotumor cerebri.

Reproductive Healthcare

Main article: Transgender pregnancy

There are frequent misconceptions within both patients and doctors about how hormone replacement therapy affects fertility. One common misconception is that starting it automatically leads to infertility. While it may impact the ability to be fertile, it does not mean it leads to a hundred percent infertility rate. There have been numerous cases of transgender men experiencing pregnancy and abortion. As trans men and doctors can be under this misconception about hormone replacement therapy impacting fertility and serving as a form of contraception, keeping people informed on fertility options remains crucial.

For trans women, it is possible for them to undergo cryopreservation before starting hormone replacement therapy. As evidence has shown that trans women tend to have lower motile sperm compared to their cisgender counterparts, fertility preservation can be important for individuals anticipating having biological children in the future. While fertility preservation is important to consider before starting HRT, it is possible in some cases to regain fertility after halting HRT for a period of time.

It is also important to educate transgender youth on their fertility preservation options. This is because few adolescents end up doing so, alongside transgender adolescents reporting distress at the prospect of becoming infertile due to medical conditions and treatment relating to their transgender identity.

Gender-affirming surgery

Main article: gender-affirming surgery

The goal of gender-affirming surgery is to align the secondary sexual characteristics of transgender people with their gender identity. As hormone replacement therapy, gender-affirming surgery is also employed as a response to diagnosis gender dysphoria.

The World Professional Association for Transgender Health (WPATH) Standards of Care recommend additional requirements for gender-affirming surgery when compared to hormone replacement therapy. Whereas hormone replacement therapy can be obtained through something as simple as an informed consent form, gender-affirming surgery can require a supporting letter from a licensed therapist (two letters for genital surgery such as vaginoplasty or phalloplasty), hormonal treatment, and (for genital surgery) completion of a 12-month period in which the person lives full-time as their gender. WPATH standards, while commonly used in gender clinics, are non-binding; many trans patients undergoing surgery do not meet all of the eligibility criteria.

Effectiveness

The need for treatment is emphasized by the higher rate of mental health problems, including depression, anxiety, and various addictions, as well as a higher suicide rate among untreated trans people than in the general population. Many of these problems, in the majority of cases, disappear or decrease significantly after social and/or medical transition.

Medical transition in the form of HRT and/or gender-affirming surgery typically results in improved quality of life. After treatment, most trans people experience improved psychological, social, and sexual functioning, and improved global functioning. Less than 1% of post-operative trans patients regret surgery.

Gender-affirming surgery alone may not be sufficient treatment in all cases. Dysphoria and thus suicidality may persist, and some trans people may need further health care in addition to surgery.

In 2012, the American Psychiatric Association and Royal College of Psychiatrists concluded that certain robust study methods, such as a randomized controlled trial, cannot be carried out on most aspects of transgender health care (especially surgery) due to the nature of the treatment itself: It's not ethical to randomly select study participants for vaginoplasty, for example.

Issues affecting transgender patients

Violence

The heightened levels of violence and abuse that transgender people experience result in unique adverse effects on bodily and mental health. Specifically, in resource-constrained settings where non-discriminatory policies may be limited or not enforced, transgender people may encounter high rates of stigma and violence which are associated with poor health outcomes. Studies in countries of the Global North show higher levels of discrimination and harassment in school, workplace, healthcare services and the family when compared with cisgender populations, situating transphobia as a key health risk factor for the physical and mental health of transgender people.

There is limited data regarding the impact of social determinants of health on transgender and gender non-conforming individuals health outcomes. However, despite the limited data available, transgender and gender non-conforming individuals have been found to be at higher risk of experiencing poor health outcomes and restricted access to health care due to increased risk for violence, isolation, and other types of discrimination both inside and outside the health care setting.

Despite its importance, access to preventive care is also limited by several factors, including discrimination and erasure. A study on young transgender women's access to HIV treatment found that one of the main contributors to not accessing care was the use of incorrect name and pronouns. A meta analysis of the National Transgender Discrimination Survey examined respondents who used the "gender not listed here" option on the survey and their experiences with accessing health care. Over a third of the people who chose that option said that they had avoided accessing general care due to bias and fears of social repercussions.

Mental health

Transgender individuals may experience distress and sadness as a result of their gender identity being inconsistent with their biological sex. This distress is referred to as gender dysphoria. Gender dysphoria is typically most upsetting for the individual prior to transitioning, and once the individual begins to transition into their desired gender, whether the transition be socially, medically, or both, the distress frequently lessens.

Those who are transgender are significantly more likely to be diagnosed with anxiety disorders or depression than the general population. A number of studies suggest that the inflated rates of depression and anxiety in transgender individuals may partially be because of systematic discrimination or a lack of support. Evidence suggests that these increased rates begin to normalize when transgender individuals are accepted as their identified gender and when they live within a supportive household.

Many studies report extremely high rates of suicide within the transgender community. A United States study of 6,450 transgender individuals found that 41% of them had attempted suicide, as differing from the national average of 4.6%. The very same survey found that these rates were the most high for certain demographics, with transgender youth between the ages of 18 and 24 having the highest percent. Individuals in the survey who were multiracial, had lower levels of education, and those with a lower annual income were all more likely to have attempted. Specifically, transgender males as a group are the most likely to attempt suicide, more so than transgender females. Later surveys suggest that the rate of suicidal attempts for non-binary individuals is in between the two. Transgender adults who have "de-transitioned", meaning having gone back to living as their sex assigned at birth, are significantly more likely to attempt suicide than transgender adults who have never "de-transitioned".

Several studies have shown the relation between minority stress and the heightened rate of depression and other mental illness among both transgender men and women. The expectation to experience rejection can become an important stressor for transgender and gender non-conforming individuals. Mental health problems among trans people are related to higher rates of self-harm, drug usage, and suicidal ideations and attempts.

Health experiences

Trans people are a vulnerable population of patients with negative experiences in health care contributing to stigmatization of their gender identity. As noted by a systematic review conducted by researchers at James Cook University, evidence reports that 75.3% of respondents have negative experiences during physician visits when seeking gender identity-based care.

Clinical environment

Guidelines from the UCSF Transgender Care Center state the importance of visibility in chosen gender identity for transgender or non-binary patients. Safe environments include a two-step process in collecting gender identity data by differentiating between personal identity and assignments at birth for medical histories. Common techniques recommended are asking patients their preferred name, pronouns, and other names they may go by in legal documents. In addition, visibility of non-cisgender identities is defined by the work environment of the clinic. Front-desk staff and medical assistants will interact with patients, which these guidelines recommend appropriate training. The existence of at least one gender-neutral bathroom shows consideration of patients with non-binary gender identities.

Clinicians may improperly connect transgender people's symptoms to their gender transition, a phenomenon known as trans broken arm syndrome. Trans broken arm syndrome is particularly prevalent among mental health practitioners, but it exists in all fields of medicine. Misguided investigation of transition-related causes can frustrate patients and cause delay in or refusal of treatment, or misdiagnosis and prescription of a wrong treatment. Misattribution of symptoms to transgender hormone therapy may also cause doctors to erroneously recommend the patient stop taking hormones. Trans broken arm syndrome may also manifest as health insurance companies refusing to pay for treatments, claiming the treated condition is caused by the patient's transgender status, and thus is a pre-existing condition. According to The SAGE Encyclopedia of Trans Studies, trans broken arm syndrome is a form of discrimination against transgender people. A 2021 survey by TransActual shows that 57% of transgender people in the United Kingdom put off seeing a doctor when they were ill. In 2014, 43% of transgender counselling clients in the UK said their counsellor "wanted to explore transgender issues in therapy even when this wasn't the reason they had sought help".

Global access

Global access to healthcare across primary and secondary health settings remains fragmented for transgender people, with access and services highly dependent on a political administration's support for trans health in policy as well as globally-engrained health inequalities largely shaped by financial wealth inequalities such as the Global North and Global South divide.

Africa

South Africa

Access to transition care, mental care, and other issues affecting transgender people is very limited; there is only one comprehensive transgender health care clinic available in South Africa. Additionally, the typical lack of access to transition options that comes as a result of gatekeeping is compounded by the relatively limited knowledge of transgender topics among psychiatrists and psychologists in South Africa.

Asia

Thailand

Transgender women, known as kathoeys, have access to hormones through non-prescription sources. This kind of access is a result of the low availability and expense of transgender health care clinics. However, transgender men have difficulty gaining access to hormones such as testosterone in Thailand because it is not as readily available as hormones for kathoeys. As a result, just a third of all trans men surveyed are taking hormones to transition whereas almost three quarters of kathoeys surveyed are taking hormones.

Mainland China

A 2017 report conducted by Beijing LGBT Center and Peking University out of 1279 of its respondents who wanted to receive hormone treatment, 71% of them felt that it was "difficult", "very difficult", or "virtually impossible" to acquire safe and reliable information about gender affirming medications and receive hormonal replacement therapy with the guidance of a doctor. As a result, 66% of the respondents chose "online" and 51% chose "friends" as one of their sources for hormone replacement therapy medications. Gender reassignment surgeries were reported to be similarly inaccessible, with 89.1% of the respondents who have the needs for such surgeries unable to pursue them.

On December 1, the Chinese National Medical Products Administration banned online sales of cyproterone acetate, estradiol, and testosterone.

Europe

Spain

Public health care services are available for transgender individuals in Spain, although there has been debate over whether certain procedures should be covered under the public system. The region of Andalusia was the first to approve sex reassignment procedures, including sex reassignment surgery and mastectomies, in 1999, and several other regions have followed their lead in the following years. Multiple interdisciplinary clinics exist in Spain to cater specifically to diagnosing and treating transgender patients, including the Andalusian Gender Team. As of 2013, over 4000 transgender patients had been treated in Spain, including Spaniards and international patients.

Beginning in 2007, Spain has begun allowing transgender individuals who are eighteen years or older to change their name and gender identity on public records and documents if they have been receiving hormone replacement therapy for at least two years.

Sweden

In 1972, Sweden introduced a law that made it possible to change a person's legal gender, but in order to do that, transgender individuals were required to be sterilized and were not allowed to save any sperm or eggs. Apart from this, there were no other mandatory surgeries required for legal gender change. In 1999, people who had been forcibly sterilized in Sweden were entitled to compensation. However, the sterilization requirement remained for people who changed their legal gender. In January 2013, forced sterilization was banned in Sweden.

Depending on the persons health and wishes there are a number of different treatments and surgeries available. Today, no form of treatment is mandatory. An individual with a transsexual or gender dysphoria diagnosis can, together with the assessment team and other doctors, decide what suits them. Although, in order to access medical and legal transitional treatment (e.g. hormone replacement therapy, and top surgery to enhance or remove breast tissue), the person will need to be diagnosed with transexualism or gender dysphoria, which requires at least one year of therapy. Medically transitioning can be very expensive, but in Sweden, the whole treatment is covered by the high-cost protection for medications and doctor's visits and there is no surgery fee. The fee the individual pays for a doctor's appointment or other care represents only a small fraction of the actual costs. If a person would like to change their legal gender marker and personal identity number they will have to seek permission from the National Board of Health and Welfare. For non-binary persons younger than 18 years, the healthcare is very limited. These individuals do not have access to a legal gender marker change or bottom surgery.

In Sweden, anyone is allowed to change their name at any time, including for gender transition.

Up until January 27, 2017, being transsexual was classed as a disease. Two months earlier, on November 21, 2016, around 50 trans activists broke into and occupied the Swedish National Board of Health and Welfare (Swedish: Socialstyrelsen) premises in Rålambsvägen in Stockholm. The activists demanded that their voices be heard regarding the way the country, healthcare, and the National Board of Health and Welfare mistreat transgender and intersex individuals.

Netherlands

A sign at a rally calling for equal access to health care for transgender people

Gender care in the Netherlands is insured under the national health care of third part insurer's, including laser hair removal, SRS, facial feminization surgery and hormones. Hormones can be prescribed by licensed endocrinologist in an academic hospital from the age 16 and older. Blockers can be prescribed from age 12 when puberty usually starts.

United Kingdom

In 2018 Stonewall described UK transgender healthcare as having "significant barriers to accessing treatment, including waiting times that stretch into years, far exceeding the maximums set by law for NHS patients". Patients have the legal right to begin treatment within 18 weeks of referral by their GP, however the average wait for patients to gender identity clinics was 18 months in 2020 with over 13,000 people on the waiting list for appointments at gender identity clinics. 

A 2013 survey of gender identity clinic services provided by the UK National Health Service (NHS) found that 94% of transgender people using the gender identity clinics were satisfied with their care and would recommend the clinics to a friend or family member. This study focused on transgender people using the NHS clinics and so was prone to survivorship bias, as those unhappy with the NHS service are less likely to use it. Despite this positive response, however, other National Health Service programs are lacking; almost a third of respondents reported inadequate psychiatric care in their local area. The options available from the National Health Service also vary with location; slightly differing protocols are used in England, Scotland, Wales and Northern Ireland. Protocols and available options differ widely outside of the UK.

Scotland

There are four NHS Scotland Gender Identity Clinics providing services to adults and a separate service for younger people. The National Gender Identity Clinical Network for Scotland reported in 2021 that some patients had waited in excess of two years from referral for their first appointment. Minister for Public Health Maree Todd has stated that the Scottish Government wants to reduce "unacceptable waits to access gender identity services". Research has indicated patient dissatisfaction with long wait times. However, overall experience of treatment outcomes was largely positive, particularly for hormone therapy and surgery.

North America

Canada

A study of transgender Ontario residents aged 16 and over, published in 2016, found that half of them were reluctant to discuss transgender issues with their family doctor. A 2013–2014 nationwide study of young transgender and genderqueer Canadians found that a third of younger (ages 14–18) and half of the older (ages 19–25) respondents missed needed physical health care. Only 15 percent of respondents with a family doctor felt very comfortable discussing transgender issues with them.

All Canadian provinces fund some sex reassignment surgeries, with New Brunswick being the last of the provinces to start insuring these procedures in 2016. Waiting times for surgeries can be lengthy, as few surgeons in the country provide them; a clinic in Montreal is the only one providing a full range of procedures. Insurance coverage is not generally provided for the transition-related procedures of facial feminization surgery, tracheal shave, or laser hair removal.

Blood donation

Canada's blood collection organization Canadian Blood Services has eligibility criteria for transgender people, which came into effect on August 15, 2016. This criteria states that transgender donors who have not had lower gender affirming surgery will be asked questions based on their sex assigned at birth. They will be eligible to donate or be deferred based on these criteria. For example, trans women will be asked if they have had sex with a man in the last 3 months. If the response is yes, they will be deferred for 3 months after their last sexual contact with a man. Donors who have had lower gender affirming surgery will be deferred from donating blood for 3 months after their surgery. After those months, these donors will be screened in their affirmed gender.

Mexico

A July 2016 study in The Lancet Psychiatry reported that nearly half of transgender people surveyed undertook body-altering procedures without medical supervision. Transition-related care is not covered under Mexico's national health plan. Only one public health institution in Mexico provides free hormones for transgender people. Health care for transgender Mexicans focuses on HIV and prevention of other sexually transmitted diseases.

The Lancet study also found that many transgender Mexicans have physical health problems due to living on the margins of society. The authors of the study recommended that the World Health Organization declassify transgender identity as a mental disorder, to reduce stigma against this population.

United States

See also: Transgender rights in the United States § Healthcare

Transgender people face various kinds of discrimination, especially in health care situations. An assessment of transgender needs in Philadelphia found that 26% of respondents had been denied health care because they were transgender and 52% of respondents had difficulty accessing health services. Aside from transition related care, transgender and gender non-conforming individuals need preventive care such as vaccines, gynecological care, prostate exams, and other annual preventive health measures. Various factors play a role in creating the limited access to care, such as insurance coverage issues related to their legal gender identity status.

The Affordable Care Act (commonly known as Obamacare) marketplace has improved access to insurance for the LGBT community through anti-discriminatory measures, such as not allowing insurance companies to reject consumers for being transgender. However, insurance sold outside of the ACA marketplace does not have to follow these requirements. This means that preventive care, such as gynecological exams for transgender men, may not be covered.

South America

Colombia

Transgender women sex workers have cited financial difficulties as barriers to accessing physical transition options. As a result, they have entered sex work to relieve financial burdens, both those related to transition and those not related to transition. However, despite working in the sex trade, the transgender women are at low risk for HIV transmission as the Colombian government requires education about sexual health and human rights for sex workers to work in so-called tolerance zones, areas where sex work is legal.

For transgender youth

See also: Suicide among LGBT youth

Transition options for transgender adolescents and youth are significantly limited compared to those for transgender adults. Prepubescent transgender youth can go through various social changes, such as presenting as their gender and asking to be called by a different name or different pronouns. Medical options for transition become available once the child begins to enter puberty. Under close supervision by a team of doctors, puberty blockers may be used to limit the effects of puberty.

Discrimination has a significant effect on the mental health of young transgender people. The lack of family acceptance, rejection in schools and abuse from peers can be powerful stressors, leading to poor mental health and substance abuse. A study done on transgender youth in San Francisco found that higher rates of both transgender-based and racial bias are associated with increased rates of depression, post-traumatic stress disorder, and suicidal ideation.

In a 2018 review, evidence suggested that hormonal treatments for transgender adolescents can achieve their intended physical effects. The mental effects of GnRH modifiers are positive with treatment associated with significant improvements in multiple psychological measures, including global functioning, depression, and overall behavioral and/or emotional problems. In a two-year study published in January 2023, Chen et al. found that gender-affirming hormones for transgender and non-binary youth "improved appearance congruence and psychosocial functioning". Another study analyzing Dutch transgender youth completed by Catharina van der Loos et al. found that 98% of participants who started gender-affirming hormone treatment in youth continued using said treatment into adulthood.

For transgender older adults

Transgender older adults can encounter challenges in the access and quality of care received in health care systems and nursing homes, where providers may be ill-prepared to provide culturally sensitive care to trans people. Trans individuals face the risk of aging with more limited support and in more stigmatizing environments than heteronormative individuals. Despite the rather negative picture portrayed by medical literature in relation to the depression and isolation that many transgender people encounter at earlier stages of life, some studies found testimonies of older LGBT adults relating feelings of inclusion, comfort and community support.