Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 4

Use $\hslash =c=1$ and the mostly-plus Minkowski metric. Then the source's Klein-Gordon equation is $({\partial }_t^2-{\nabla }^2+m^2)\varphi =0$. For a plane wave $\varphi =C{e}^{-i\omega t+i\mathbf{k}\cdot \mathbf{x}}$, substitution givesThus a real propagating wave vector requires $|\omega |\ge m$, or $|E|\ge m{c}^2$ on restoring units. The positive-energy branch has a rest-energy threshold $m{c}^2$. At $|\omega |=m$, $\mathbf{k}=0$; frequencies below the mass threshold give evanescent spatial dependence rather than real-wave-vector propagation. Both signs of $\omega$ occur in the classical equation; a real scalar field combines complex-conjugate modes. Its group velocity is $\mathbf{k}/\omega$ and is subluminal, though the positive-branch phase speed exceeds one when $m>0$.

For a static spherically symmetric scalar field, the equation outside the source isSetting $\psi =r\varphi$ gives ${\psi }^{\prime \prime }-m^2\psi =0$, soDecay at infinity selects $B=0$, the Yukawa potential. It is singular at the origin and describes a point source distributionally; a source-free solution regular at the origin and decaying at infinity must be zero. For $m=0$, $\varphi =A/r+B$, and decay sets the constant $B$ to zero.

Restoring units, the decaying Yukawa potential is $V(r)=C{e}^{-r/{\lambda }_m}/r$ with range ${\lambda }_m=\hslash /(mc)$. For $r_{\ast }=10^{-18}\mathrm{m}$,The geometric $1/r$ factor belongs in the potential ratio; the exponential alone is the additional suppression relative to massless exchange. Using $\hslash c=197.32698\mathrm{MeV}\mathrm{fm}$ givesThe following are ratios of potential magnitudes to the magnitude at $r_{\ast }$, with the same coupling held fixed:

For example, the pion's nuclear exponential factor is $e^{-0.68346}=0.505$, whereas the Higgs nuclear exponent is about $-632.83$. The huge atomic suppressions are most reliably reported through logarithms, not through floating-point exponentiation that silently underflows to zero. These are potential energy ratios, not force ratios or absolute comparisons of different interaction strengths.

A pion mass gives a Yukawa potential range of order a femtometre, so meson exchange can describe the short range of the residual nuclear force. The nuclear suppression relative to a massless potential of the same coupling is only about a factor of two at one femtometre. A Higgs-mass mediator instead has a range around $10^{-18}\mathrm{m}$ and a further factor $e^{-633}$ by nuclear separations; direct Higgs exchange cannot supply an appreciable long-range nuclear binding interaction at those separations. The range comparison does not replace the spin, coupling and many-body physics of actual nuclear forces.

At atomic separations both massive exchange potentials are exponentially negligible. Molecular binding needs an interaction that survives at atomic distances; actual chemical bonds is governed principally by electromagnetism and quantum mechanics, with a massless photon rather than a scalar mediator. A hypothetical massless scalar would also have an unscreened $1/r$ potential, but its coupling and spin-dependent phenomenology would be different. The Higgs mechanism affects particle masses; that does not imply an appreciable direct Higgs-exchange force between atoms separated by $10^{-10}\mathrm{m}$.

A relativistic two-component spinor is an element of a two-dimensional complex representation of the special linear group $SL(2,\mathbb{C})$, the double cover of the proper orthochronous Lorentz group. A Weyl spinor transforms in either $(1/2,0)$ or $(0,1/2)$; parity interchanges the two chiralities. It describes a chiral spin-$1/2$ degree of freedom. A free massless Weyl field has one particle helicity and the opposite antiparticle helicity.

The number of components alone does not prove zero mass. A massive Dirac field has two coupled two-component chiral fields, and a nonrelativistic Pauli spinor describes the two spin states of a massive spin-$1/2$ particle. A Majorana mass term can also be written using a two-component field and its complex conjugate. Thus a single independent free massless Weyl equation is a dynamical assumption, not part of the bare algebraic definition of a two-component spinor.

Write $u=(a,b{)}^T$. The Pauli matrices giveHenceTherefore $\boxed{U_{\mu }{U}^{\mu }=0}$, with $U^0>0$ for $u\ne 0$: the bilinear is a future null Lorentz four-vector, and $u=0$ gives the zero vector.

For $S=e^{\rho {\sigma }_z/2}=\mathrm{diag}(e^{\rho /2},e^{-\rho /2})$, the transformed components are $a^{\prime }=e^{\rho /2}a$ and $b^{\prime }=e^{-\rho /2}b$. Thus $a^{\prime \ast }{b}^{\prime }=a^{\ast }b$, while the two squared magnitudes acquire factors $e^{\rho }$ and $e^{-\rho }$. Substitution givesThe matrix preserves the Minkowski metric and is an active boost along $+z$ of rapidity $\rho$, with velocity $c\tanh \rho$. Relative to the passive standard-boost convention of part i, its velocity parameter has the opposite sign.

For $S=e^{i\theta {\sigma }_z}$, $a^{\prime }=e^{i\theta }a$ and $b^{\prime }=e^{-i\theta }b$, so $a^{\prime \ast }{b}^{\prime }=e^{-2i\theta }{a}^{\ast }b$. ConsequentlyThis is an active right-handed spatial rotation through $-2\theta$ about $z$. The minus sign follows from the specified positive $i\theta {\sigma }_z$ exponent; the familiar positive-angle spin rotation uses $e^{-i\phi {\sigma }_z/2}$. The doubled angle exhibits the Lorentz spinor double cover: $u$ and $-u$ give the same four-vector.

For a free massless Weyl spinor, yes: each component obeys the massless Klein-Gordon equation. In $c=\hslash =1$ units, take either chiral Weyl equation, for exampleApply its opposite-sign partner. The Pauli matrix multiplication identity makes the antisymmetric part vanish against the symmetric pair of spatial derivatives, soThus $\boxed{{\partial }_{\mu }{\partial }^{\mu }u=0}$ componentwise. This does not turn the field into a scalar: its components still mix under a Spinor representation of the Lorentz group. The second-order equation is a necessary consequence of the first-order equation, but does not by itself impose the chiral helicity relation.

There is no unconditional massless-wave conclusion from merely being a two-component spinor, or from the null bilinear of part d. For example, $u(t,\mathbf{x})=e^{-i\omega t}(1,0{)}^T$ has a null bilinear at every point, yet $({\partial }_t^2-{\nabla }^2)u=-{\omega }^{2}u\ne 0$ for $\omega \ne 0$. Massive free two-component chiral fields coupled through a mass term instead obey the massive Klein-Gordon equation; interacting fields can have additional terms. The affirmative result therefore uses the free, massless dynamics.