Relativistic quantum mechanics

Spin and electromagnetically interacting particles

Including interactions in RWEs is generally difficult. Minimal coupling is a simple way to include the electromagnetic interaction. For one charged particle of electric charge q in an electromagnetic field, given by the magnetic vector potential A(r, t) defined by the magnetic field B = ∇ × A, and electric scalar potential ϕ(r, t), this is:

${\displaystyle \hat{E}\to \hat{E}-q\varphi ,\hat{\mathbf{p}}\to \hat{\mathbf{p}}-q\mathbf{A}\rightleftharpoons {\hat{P}}_{\mu }\to {\hat{P}}_{\mu }-q{A}_{\mu }}$

where P_μ is the four-momentum that has a corresponding 4-momentum operator, and A_μ the four-potential. In the following, the non-relativistic limit refers to the limiting cases:

${\displaystyle E-e\varphi \approx m{c}^2,\mathbf{p}\approx m\mathbf{v},}$

that is, the total energy of the particle is approximately the rest energy for small electric potentials, and the momentum is approximately the classical momentum.

Spin 0

In RQM, the KG equation admits the minimal coupling prescription;

${\displaystyle {(\hat{E}-q\varphi )}^2\psi =c^2{(\hat{\mathbf{p}}-q\mathbf{A})}^2\psi +(m{c}^2{)}^2\psi \rightleftharpoons \left[({\hat{P}}_{\mu }-q{A}_{\mu })({\hat{P}}^{\mu }-q{A}^{\mu })-{(mc)}^2\right]\psi =0.}$

In the case where the charge is zero, the equation reduces trivially to the free KG equation so nonzero charge is assumed below. This is a scalar equation that is invariant under the irreducible one-dimensional scalar (0,0) representation of the Lorentz group. This means that all of its solutions will belong to a direct sum of (0,0) representations. Solutions that do not belong to the irreducible (0,0) representation will have two or more independent components. Such solutions cannot in general describe particles with nonzero spin since spin components are not independent. Other constraint will have to be imposed for that, e.g. the Dirac equation for spin ⁠1/2⁠, see below. Thus if a system satisfies the KG equation only, it can only be interpreted as a system with zero spin.

The electromagnetic field is treated classically according to Maxwell's equations and the particle is described by a wavefunction, the solution to the KG equation. The equation is, as it stands, not always very useful, because massive spinless particles, such as the π-mesons, experience the much stronger strong interaction in addition to the electromagnetic interaction. It does, however, correctly describe charged spinless bosons in the absence of other interactions.

The KG equation is applicable to spinless charged bosons in an external electromagnetic potential. As such, the equation cannot be applied to the description of atoms, since the electron is a spin ⁠1/2⁠ particle. In the non-relativistic limit the equation reduces to the Schrödinger equation for a spinless charged particle in an electromagnetic field:

${\displaystyle \left(i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}-q\varphi \right)\psi =\frac{1}{2m}{(\hat{\mathbf{p}}-q\mathbf{A})}^2\psi \iff \hat{H}=\frac{1}{2m}{(\hat{\mathbf{p}}-q\mathbf{A})}^2+q\varphi .}$

Spin ⁠1/2⁠

Non relativistically, spin was phenomenologically introduced in the Pauli equation by Pauli in 1927 for particles in an electromagnetic field:

${\displaystyle \left(i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}-q\varphi \right)\psi =\left[\frac{1}{2m}{(\mathit{\sigma }\cdot (\mathbf{p}-q\mathbf{A}))}^2\right]\psi \iff \hat{H}=\frac{1}{2m}{(\mathit{\sigma }\cdot (\mathbf{p}-q\mathbf{A}))}^2+q\varphi }$

by means of the 2 × 2 Pauli matrices, and ψ is not just a scalar wavefunction as in the non-relativistic Schrödinger equation, but a two-component spinor field:

${\displaystyle \psi =\left(\begin{array}{c}{\psi }_{\uparrow }\\ {\psi }_{\downarrow }\end{array}\right)}$

where the subscripts ↑ and ↓ refer to the "spin up" (σ = +⁠1/2⁠) and "spin down" (σ = −⁠1/2⁠) states.

In RQM, the Dirac equation can also incorporate minimal coupling, rewritten from above;

${\displaystyle \left(i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}-q\varphi \right)\psi ={\gamma }^0\left[c\mathit{\gamma }\cdot (\hat{\mathbf{p}}-q\mathbf{A})-m{c}^2\right]\psi \rightleftharpoons \left[{\gamma }^{\mu }({\hat{P}}_{\mu }-q{A}_{\mu })-m{c}^2\right]\psi =0}$

and was the first equation to accurately predict spin, a consequence of the 4 × 4 gamma matrices γ⁰ = β, γ = (γ₁, γ₂, γ₃) = βα = (βα₁, βα₂, βα₃). There is a 4 × 4 identity matrix pre-multiplying the energy operator (including the potential energy term), conventionally not written for simplicity and clarity (i.e. treated like the number 1). Here ψ is a four-component spinor field, which is conventionally split into two two-component spinors in the form:

${\displaystyle \psi =\left(\begin{array}{c}{\psi }_{+}\\ {\psi }_{-}\end{array}\right)=\left(\begin{array}{c}{\psi }_{+\uparrow }\\ {\psi }_{+\downarrow }\\ {\psi }_{-\uparrow }\\ {\psi }_{-\downarrow }\end{array}\right)}$

The 2-spinor ψ₊ corresponds to a particle with 4-momentum (E, p) and charge q and two spin states (σ = ±⁠1/2⁠, as before). The other 2-spinor ψ₋ corresponds to a similar particle with the same mass and spin states, but negative 4-momentum −(E, p) and negative charge −q, that is, negative energy states, time-reversed momentum, and negated charge. This was the first interpretation and prediction of a particle and corresponding antiparticle. See Dirac spinor for further description of these spinors. In the non-relativistic limit the Dirac equation reduces to the Pauli equation (see Dirac equation for how). When applied a one-electron atom or ion, setting A = 0 and ϕ to the appropriate electrostatic potential, additional relativistic terms include the spin–orbit interaction, electron gyromagnetic ratio, and Darwin term. In ordinary QM these terms have to be put in by hand and treated using perturbation theory. The positive energies do account accurately for the fine structure.

Within RQM, for massless particles the Dirac equation reduces to:

${\displaystyle \left(\frac{\hat{E}}{c}+\mathit{\sigma }\cdot \hat{\mathbf{p}}\right){\psi }_{+}=0,\left(\frac{\hat{E}}{c}-\mathit{\sigma }\cdot \hat{\mathbf{p}}\right){\psi }_{-}=0\rightleftharpoons {\sigma }^{\mu }{\hat{P}}_{\mu }{\psi }_{+}=0,{\sigma }_{\mu }{\hat{P}}^{\mu }{\psi }_{-}=0,}$

the first of which is the Weyl equation, a considerable simplification applicable for massless neutrinos. This time there is a 2 × 2 identity matrix pre-multiplying the energy operator conventionally not written. In RQM it is useful to take this as the zeroth Pauli matrix σ₀ which couples to the energy operator (time derivative), just as the other three matrices couple to the momentum operator (spatial derivatives).

The Pauli and gamma matrices were introduced here, in theoretical physics, rather than pure mathematics itself. They have applications to quaternions and to the SO(2) and SO(3) Lie groups, because they satisfy the important commutator [ , ] and anticommutator [ , ]₊ relations respectively:

${\displaystyle \left[{\sigma }_a,{\sigma }_b\right]=2i{\epsilon }_{abc}{\sigma }_c,{\left[{\sigma }_a,{\sigma }_b\right]}_{+}=2{\delta }_{ab}{\sigma }_0}$

where ε_abc is the three-dimensional Levi-Civita symbol. The gamma matrices form bases in Clifford algebra, and have a connection to the components of the flat spacetime Minkowski metric η^αβ in the anticommutation relation:

${\displaystyle {\left[{\gamma }^{\alpha },{\gamma }^{\beta }\right]}_{+}={\gamma }^{\alpha }{\gamma }^{\beta }+{\gamma }^{\beta }{\gamma }^{\alpha }=2{\eta }^{\alpha \beta },}$

(This can be extended to curved spacetime by introducing vierbeins, but is not the subject of special relativity).

In 1929, the Breit equation was found to describe two or more electromagnetically interacting massive spin ⁠1/2⁠ fermions to first-order relativistic corrections; one of the first attempts to describe such a relativistic quantum many-particle system. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums.

Helicity and chirality

The helicity operator is defined by;

${\displaystyle \hat{h}=\hat{\mathbf{S}}\cdot \frac{\hat{\mathbf{p}}}{|\mathbf{p}|}=\hat{\mathbf{S}}\cdot \frac{c\hat{\mathbf{p}}}{\sqrt{E^2-(m_0{c}^2{)}^2}}}$

where p is the momentum operator, S the spin operator for a particle of spin s, E is the total energy of the particle, and m₀ its rest mass. Helicity indicates the orientations of the spin and translational momentum vectors. Helicity is frame-dependent because of the 3-momentum in the definition, and is quantized due to spin quantization, which has discrete positive values for parallel alignment, and negative values for antiparallel alignment.

An automatic occurrence in the Dirac equation (and the Weyl equation) is the projection of the spin ⁠1/2⁠ operator on the 3-momentum (times c), σ · c p, which is the helicity (for the spin ⁠1/2⁠ case) times ${\displaystyle \sqrt{E^2-(m_0{c}^2{)}^2}}$.

For massless particles the helicity simplifies to:

${\displaystyle \hat{h}=\hat{\mathbf{S}}\cdot \frac{c\hat{\mathbf{p}}}{E}}$

Higher spins

The Dirac equation can only describe particles of spin ⁠1/2⁠. Beyond the Dirac equation, RWEs have been applied to free particles of various spins. In 1936, Dirac extended his equation to all fermions, three years later Fierz and Pauli rederived the same equation. The Bargmann–Wigner equations were found in 1948 using Lorentz group theory, applicable for all free particles with any spin. Considering the factorization of the KG equation above, and more rigorously by Lorentz group theory, it becomes apparent to introduce spin in the form of matrices.

The wavefunctions are multicomponent spinor fields, which can be represented as column vectors of functions of space and time:

${\displaystyle \psi (\mathbf{r},t)=\left[\begin{array}{c}{\psi }_{\sigma =s}(\mathbf{r},t)\\ {\psi }_{\sigma =s-1}(\mathbf{r},t)\\ ⋮\\ {\psi }_{\sigma =-s+1}(\mathbf{r},t)\\ {\psi }_{\sigma =-s}(\mathbf{r},t)\end{array}\right]\rightleftharpoons {\psi (\mathbf{r},t)}^{†}=\left[\begin{array}{ccccc}{{\psi }_{\sigma =s}(\mathbf{r},t)}^{\star }& {{\psi }_{\sigma =s-1}(\mathbf{r},t)}^{\star }& \cdots & {{\psi }_{\sigma =-s+1}(\mathbf{r},t)}^{\star }& {{\psi }_{\sigma =-s}(\mathbf{r},t)}^{\star }\end{array}\right]}$

where the expression on the right is the Hermitian conjugate. For a massive particle of spin s, there are 2s + 1 components for the particle, and another 2s + 1 for the corresponding antiparticle (there are 2s + 1 possible σ values in each case), altogether forming a 2(2s + 1)-component spinor field:

${\displaystyle \psi (\mathbf{r},t)=\left[\begin{array}{c}{\psi }_{+,\sigma =s}(\mathbf{r},t)\\ {\psi }_{+,\sigma =s-1}(\mathbf{r},t)\\ ⋮\\ {\psi }_{+,\sigma =-s+1}(\mathbf{r},t)\\ {\psi }_{+,\sigma =-s}(\mathbf{r},t)\\ {\psi }_{-,\sigma =s}(\mathbf{r},t)\\ {\psi }_{-,\sigma =s-1}(\mathbf{r},t)\\ ⋮\\ {\psi }_{-,\sigma =-s+1}(\mathbf{r},t)\\ {\psi }_{-,\sigma =-s}(\mathbf{r},t)\end{array}\right]\rightleftharpoons {\psi (\mathbf{r},t)}^{†}\left[\begin{array}{cccc}{{\psi }_{+,\sigma =s}(\mathbf{r},t)}^{\star }& {{\psi }_{+,\sigma =s-1}(\mathbf{r},t)}^{\star }& \cdots & {{\psi }_{-,\sigma =-s}(\mathbf{r},t)}^{\star }\end{array}\right]}$

with the + subscript indicating the particle and − subscript for the antiparticle. However, for massless particles of spin s, there are only ever two-component spinor fields; one is for the particle in one helicity state corresponding to +s and the other for the antiparticle in the opposite helicity state corresponding to −s:

${\displaystyle \psi (\mathbf{r},t)=\left(\begin{array}{c}{\psi }_{+}(\mathbf{r},t)\\ {\psi }_{-}(\mathbf{r},t)\end{array}\right)}$

According to the relativistic energy-momentum relation, all massless particles travel at the speed of light, so particles traveling at the speed of light are also described by two-component spinors. Historically, Élie Cartan found the most general form of spinors in 1913, prior to the spinors revealed in the RWEs following the year 1927.

For equations describing higher-spin particles, the inclusion of interactions is nowhere near as simple minimal coupling, they lead to incorrect predictions and self-inconsistencies. For spin greater than ⁠ħ/2⁠, the RWE is not fixed by the particle's mass, spin, and electric charge; the electromagnetic moments (electric dipole moments and magnetic dipole moments) allowed by the spin quantum number are arbitrary. (Theoretically, magnetic charge would contribute also). For example, the spin ⁠1/2⁠ case only allows a magnetic dipole, but for spin 1 particles magnetic quadrupoles and electric dipoles are also possible. For more on this topic, see multipole expansion and (for example) Cédric Lorcé (2009).

Velocity operator

The Schrödinger/Pauli velocity operator can be defined for a massive particle using the classical definition p = m v, and substituting quantum operators in the usual way:

${\displaystyle \hat{\mathbf{v}}=\frac{1}{m}\hat{\mathbf{p}}}$

which has eigenvalues that take any value. In RQM, the Dirac theory, it is:

${\displaystyle \hat{\mathbf{v}}=\frac{i}{\hslash }\left[\hat{H},\hat{\mathbf{r}}\right]}$

which must have eigenvalues between ±c. See Foldy–Wouthuysen transformation for more theoretical background.

Relativistic quantum Lagrangians

The Hamiltonian operators in the Schrödinger picture are one approach to forming the differential equations for ψ. An equivalent alternative is to determine a Lagrangian (really meaning Lagrangian density), then generate the differential equation by the field-theoretic Euler–Lagrange equation:

${\displaystyle {\mathrm{\partial }}_{\mu }\left(\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }({\mathrm{\partial }}_{\mu }\psi )}\right)-\frac{\mathrm{\partial }\mathcal{L}}{\mathrm{\partial }\psi }=0}$

For some RWEs, a Lagrangian can be found by inspection. For example, the Dirac Lagrangian is:

${\displaystyle \mathcal{L}=\overline{\psi }({\gamma }^{\mu }{P}_{\mu }-mc)\psi }$

and Klein–Gordon Lagrangian is:

${\displaystyle \mathcal{L}=-\frac{{\hslash }^2}{m}{\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }{\psi }^{\ast }{\mathrm{\partial }}_{\nu }\psi -m{c}^2{\psi }^{\ast }\psi .}$

This is not possible for all RWEs; and is one reason the Lorentz group theoretic approach is important and appealing: fundamental invariance and symmetries in space and time can be used to derive RWEs using appropriate group representations. The Lagrangian approach with field interpretation of ψ is the subject of QFT rather than RQM: Feynman's path integral formulation uses invariant Lagrangians rather than Hamiltonian operators, since the latter can become extremely complicated, see (for example) Weinberg (1995).

Relativistic quantum angular momentum

In non-relativistic QM, the angular momentum operator is formed from the classical pseudovector definition L = r × p. In RQM, the position and momentum operators are inserted directly where they appear in the orbital relativistic angular momentum tensor defined from the four-dimensional position and momentum of the particle, equivalently a bivector in the exterior algebra formalism:

${\displaystyle M^{\alpha \beta }=X^{\alpha }{P}^{\beta }-X^{\beta }{P}^{\alpha }=2X^{[\alpha }{P}^{\beta ]}\rightleftharpoons \mathbf{M}=\mathbf{X}\wedge \mathbf{P},}$

which are six components altogether: three are the non-relativistic 3-orbital angular momenta; M¹² = L³, M²³ = L¹, M³¹ = L², and the other three M⁰¹, M⁰², M⁰³ are boosts of the centre of mass of the rotating object. An additional relativistic-quantum term has to be added for particles with spin. For a particle of rest mass m, the total angular momentum tensor is:

${\displaystyle J^{\alpha \beta }=2X^{[\alpha }{P}^{\beta ]}+\frac{1}{m^2}{\epsilon }^{\alpha \beta \gamma \delta }{W}_{\gamma }{p}_{\delta }\rightleftharpoons \mathbf{J}=\mathbf{X}\wedge \mathbf{P}+\frac{1}{m^2}\star (\mathbf{W}\wedge \mathbf{P})}$

where the star denotes the Hodge star operator, and

${\displaystyle W_{\alpha }=\frac{1}{2}{\epsilon }_{\alpha \beta \gamma \delta }{M}^{\beta \gamma }{p}^{\delta }\rightleftharpoons \mathbf{W}=\star (\mathbf{M}\wedge \mathbf{P})}$

is the Pauli–Lubanski pseudovector. For more on relativistic spin, see (for example) Troshin & Tyurin (1994).

Thomas precession and spin–orbit interactions

In 1926, the Thomas precession is discovered: relativistic corrections to the spin of elementary particles with application in the spin–orbit interaction of atoms and rotation of macroscopic objects. In 1939 Wigner derived the Thomas precession.

In classical electromagnetism and special relativity, an electron moving with a velocity v through an electric field E but not a magnetic field B, will in its own frame of reference experience a Lorentz-transformed magnetic field B′:

${\displaystyle {\mathbf{B}}^{\prime }=\frac{\mathbf{E}\times \mathbf{v}}{c^2\sqrt{1-{\left(v/c\right)}^2}}.}$

In the non-relativistic limit v << c:

${\displaystyle {\mathbf{B}}^{\prime }=\frac{\mathbf{E}\times \mathbf{v}}{c^2},}$

so the non-relativistic spin interaction Hamiltonian becomes:

${\displaystyle \hat{H}=-{\mathbf{B}}^{\prime }\cdot {\hat{\mathit{\mu }}}_S=-\left(\mathbf{B}+\frac{\mathbf{E}\times \mathbf{v}}{c^2}\right)\cdot {\hat{\mathit{\mu }}}_S,}$

where the first term is already the non-relativistic magnetic moment interaction, and the second term the relativistic correction of order (v/c)², but this disagrees with experimental atomic spectra by a factor ⁠1/2⁠. It was pointed out by L. Thomas that there is a second relativistic effect: An electric field component perpendicular to the electron velocity causes an additional acceleration of the electron perpendicular to its instantaneous velocity, so the electron moves in a curved path. The electron moves in a rotating frame of reference, and this additional precession of the electron is called the Thomas precession. It can be shown that the net result of this effect is that the spin–orbit interaction is reduced by half, as if the magnetic field experienced by the electron has only one-half the value, and the relativistic correction in the Hamiltonian is:

${\displaystyle \hat{H}=-{\mathbf{B}}^{\prime }\cdot {\hat{\mathit{\mu }}}_S=-\left(\mathbf{B}+\frac{\mathbf{E}\times \mathbf{v}}{2c^2}\right)\cdot {\hat{\mathit{\mu }}}_S.}$

In the case of RQM, the factor of ⁠1/2⁠ is predicted by the Dirac equation.

History

The events which led to and established RQM, and the continuation beyond into quantum electrodynamics (QED), are summarized below [see, for example, R. Resnick and R. Eisberg (1985), and P.W Atkins (1974)]. More than half a century of experimental and theoretical research from the 1890s through to the 1950s in the new and mysterious quantum theory as it was up and coming revealed that a number of phenomena cannot be explained by QM alone. SR, found at the turn of the 20th century, was found to be a necessary component, leading to unification: RQM. Theoretical predictions and experiments mainly focused on the newly found atomic physics, nuclear physics, and particle physics; by considering spectroscopy, diffraction and scattering of particles, and the electrons and nuclei within atoms and molecules. Numerous results are attributed to the effects of spin.

Relativistic description of particles in quantum phenomena

Albert Einstein in 1905 explained of the photoelectric effect; a particle description of light as photons. In 1916, Sommerfeld explains fine structure; the splitting of the spectral lines of atoms due to first order relativistic corrections. The Compton effect of 1923 provided more evidence that special relativity does apply; in this case to a particle description of photon–electron scattering. de Broglie extends wave–particle duality to matter: the de Broglie relations, which are consistent with special relativity and quantum mechanics. By 1927, Davisson and Germer and separately G. Thomson successfully diffract electrons, providing experimental evidence of wave-particle duality.

Experiments

• 1897 J. J. Thomson discovers the electron and measures its mass-to-charge ratio. Discovery of the Zeeman effect: the splitting a spectral line into several components in the presence of a static magnetic field.
• 1908 Millikan measures the charge on the electron and finds experimental evidence of its quantization, in the oil drop experiment.
• 1911 Alpha particle scattering in the Geiger–Marsden experiment, led by Rutherford, showed that atoms possess an internal structure: the atomic nucleus.
• 1913 The Stark effect is discovered: splitting of spectral lines due to a static electric field (compare with the Zeeman effect).
• 1922 Stern–Gerlach experiment: experimental evidence of spin and its quantization.
• 1924 Stoner studies splitting of energy levels in magnetic fields.
• 1932 Experimental discovery of the neutron by Chadwick, and positrons by Anderson, confirming the theoretical prediction of positrons.
• 1958 Discovery of the Mössbauer effect: resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in a solid, useful for accurate measurements of gravitational redshift and time dilation, and in the analysis of nuclear electromagnetic moments in hyperfine interactions.

Quantum non-locality and relativistic locality

In 1935, Einstein, Rosen, Podolsky published a paper concerning quantum entanglement of particles, questioning quantum nonlocality and the apparent violation of causality upheld in SR: particles can appear to interact instantaneously at arbitrary distances. This was a misconception since information is not and cannot be transferred in the entangled states; rather the information transmission is in the process of measurement by two observers (one observer has to send a signal to the other, which cannot exceed c). QM does not violate SR. In 1959, Bohm and Aharonov published a paper on the Aharonov–Bohm effect, questioning the status of electromagnetic potentials in QM. The EM field tensor and EM 4-potential formulations are both applicable in SR, but in QM the potentials enter the Hamiltonian (see above) and influence the motion of charged particles even in regions where the fields are zero. In 1964, Bell's theorem was published in a paper on the EPR paradox, showing that QM cannot be derived from local hidden-variable theories if locality is to be maintained.

Lamb shift

In 1947, the Lamb shift was discovered: a small difference in the ²S_1⁄2 and ²P_1⁄2 levels of hydrogen, due to the interaction between the electron and vacuum. Lamb and Retherford experimentally measure stimulated radio-frequency transitions the ²S_1⁄2 and ²P_1⁄2 hydrogen levels by microwave radiation. An explanation of the Lamb shift is presented by Bethe. Papers on the effect were published in the early 1950s.

Development of quantum electrodynamics

• 1927 Dirac establishes the field of QED, also coining the term "quantum electrodynamics".
• 1943 Tomonaga begins work on renormalization, influential in QED.
• 1947 Schwinger calculates the anomalous magnetic moment of the electron. Kusch measures of the anomalous magnetic electron moment, confirming one of QED's great predictions.