# Introductory Remarks to Relativistic Quantum Mechanics

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This volume gives an introduction to incorporating special relativity in quantum mechanics. That is, described here is a theory which includes Einsein's

\[ E =mc^2 \label{1.1}\]

with Heisenberg's

\[ \Delta x \Delta p \ge \dfrac{\hbar}{2}\]

How does relativistic quantum mechanics differ from non-relativistic quantum mechanics'? That is, what new feature does special relativity bring to quantum mechanics? The answer lies in Eqution \ref{1.1} - the possibility of converting energy to mass and *vice versa.*

In nonrelativistic quantum mechanics one deals with physical systems where the total mass is fixed and invariant for all time. One solves the one-body problem, the two-body problem, and so on. And one says, for example, that the hydrogen atom consists of a proton and an electron and the deuteron consists of a proton and a neutron. These are nonrelativistic statements. These composite particles (and others) consist of these two-body configurations to be sure, but they a.lso consist of other multiparticle configurations. That is, there is a nonzero probability that the state of the hydrogen atom has components corresponding to a proton and an electron and also, for example, to a proton and an electron and any number of photons, electron-positron pairs, and pions. The state of the deuteron has similar components as do the states of all composite particles. Relativistic quantum mechanics incorporates these effects.

Relativistic effects are numerically small at the atomic and molecular levels. One can understand all of biology and chemistry, and much of physics, without incorporating special relativity into quantum mechanics.

For example, nonrelativistic quantum mechanics predicts that the lowest energy state of the free hydrogen atom is the IS1r2 state with an energy of -13.58 eV, and that the 2,91r2, 2P112, 2P372 states are degenerate with an energy of -3.40 eV. This is experimentally true at that level of accuracy. Measurement of transition energies at the pev level using high precision laser spectroscopy, however, shows that these states are not degenerate in energy: the 2Psr2 and 2,5172 levels lre 42.2rLeY and 4.38peV, respectively, above the 2P1r2 level. The splitting of the \(\ce{^2P_{1/2}}\) and \(\ce{^2S_{1/2}}\) levels is called the Lamb Shift. lt was discovered in 1947 by W.E. Lamb and R.C. Retherford.

The Dirac equation for the hydrogen atom, which was invented by P.A.M. Dirac in 1928 (and is discussed rn QLB: Some Lorentz Invariant Systems) incorporates special relativity into quantum mechanics without particle creation and annihilation. It gives an improvement over nonrelativistic quantum mechanics for the fine structure of the energy levels but it does not predict the Lamb Shift. It predicts that the 2,9112 and 2P1r2 states ale degenerate and that the \(\ce{^2P_{3/2}}\) state has a higher energy by \(45.2\,\mu eV\). Quantum electrodynamics, the *relativistic quantum field theory* of electrons and photons, which was invented in the 1930's and refined by R.P Feynman, J. Schwinger and S. Tomonaga in the 1940's (and is discussed briefly ín QLB: Relativistic Quantum Field Theory), incorporates special relativity into quantum mechanics including particle creation and annihilation. It yields the Lamb Shift and gives perfect agreement with all electron-positron-photon experiments performed to date.

Relativistic effects are not small at the subatomic level. For example, it is clearly essential to include particle creation ard annihilation effects when one tries t(r inßrpret experimental data for the reactions

\[ p+p \rightarrow d+ \pi^+ \label{1.3}\]

and

\[ p+p \rightarrow p+p+ K^+ K^- \label{1.4}\]

These are the basic reactions for the production of positive pions (Equation \ref{1.3}) at the TRIUMF accelerator and for the production of positive and negative kaon in proton-proton collisions (Equation \ref{1.4}).

Quite apart from the largeness or smallness of relativistic effects, the study of relativistic quantum mechanics forces one to consider what are the fundamental entities in terms of which one describes the physical world. Searching for these fundamental entities remains a strong human quest. The reason perhaps lies in the statement

"Once we figure out how the dice are made, we may be able to figure out who is throwing them."

The basic equations of relativistic quantum mechanics are a set of commutation relations called the Poincare Algebra. We present the Poircare Algebra in Chapter 2. The basic equations of nonrelativistic quantum mechanics are a set of commutation relations called the Galilei Algebra. We present the Galilei Algebra in Chapter 2. The Galilei Algebra is a special case of the Poincare Algebra. Lorentz invariance of a physical system is defined in Chapter 3; the Poincare Algebra is derived in Chapter 4; space inversion and time reversal are discussed in Chapter 5; and the center of mass position, center of mass velocity and internal angular momentum of a Lorentz invariant system are discussed in Chapter 6. The Appendix gives some matrices which arise in previous chapters. The volume concludes with lists of selected reference books, journal articles and theses.