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Parity preserves the Minkowski metric but reverses orientation. Hence ordinary tensor contraction makesparity even, while the Levi-Civita pseudotensor changes sign and makesparity odd. Equivalently, $C_1$ is proportional to ${\mathbf{J}}^2-{\mathbf{K}}^2$ and $C_2$ to $\mathbf{J}\cdot \mathbf{K}$. The combinations $C_1\pm i{C}_2$ are proportional to the two quadratic Lorentz Casimir invariants, and parity exchanges them exactly as it exchanges the two factors in the complexified algebra.

Write a nonzero vacuum plane electromagnetic wave with wavevector $\mathbf{k}=(\omega /c)\mathbf{s}$, where $\mathbf{s}$ is a unit propagation direction. Its propagation Lorentz four-vector is $K^{\mu }=(\omega /c,\mathbf{k})$, or, up to a scalar normalization in any one frame, $n^{\mu }=(1,\mathbf{s})$. It is null because $K_{\mu }{K}^{\mu }=0$.

Choose coordinates $x^{\mu }=(ct,\mathbf{x})$ and Minkowski metric ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$. With the supplied three-dimensional boost formulas, a consistent convention assigns the printed array to upper-index components,The index conventions for the electromagnetic field tensor then require $F_{0i}=+E_i/c$ and $F_{ij}=F^{ij}$ when both indices are lowered. The printed lower-index label therefore cannot be kept together with its electric signs and a forward propagation vector in this convention.

This is not merely an arbitrary overall sign. For a valid forward wave with $\mathbf{s}=\hat{\mathbf{z}}$, $\mathbf{E}=E_0\hat{\mathbf{x}}$ and $\mathbf{B}=(E_0/c)\hat{\mathbf{y}}$, multiplying the printed array, unchanged, by $n^{\mu }=(1,0,0,1)$ gives an $x$ component $-2E_0/c$, rather than zero. One consistent repair is to retain the printed array as $F^{\mu \nu }$ and contract with $n_{\mu }=(1,-\mathbf{s})$. Equivalently, lower both tensor indices before forming the contraction asked for in the question.

Using that corrected lower-index tensor givesTo establish both electric and magnetic transversality, include the dual electromagnetic field tensor. Choose ${\tilde{F}}^{0i}=-B_i$ and ${\tilde{F}}^{ij}={ϵ}_{ijk}{E}_k/c$; lowering both indices then givesThe source-free plane-wave Maxwell equations are precisely the pair of covariant conditionsTheir temporal components give $\mathbf{k}\cdot \mathbf{E}=0$ and $\mathbf{k}\cdot \mathbf{B}=0$. Their spatial components give the associated amplitude and orientation relations, $\mathbf{E}/c=-\mathbf{s}\times \mathbf{B}$ and $\mathbf{B}=\mathbf{s}\times \mathbf{E}/c$.

To see the Lorentz covariance explicitly, let $V_{\nu }=K^{\mu }{F}_{\mu \nu }$. It is a covector, so if $V_{\nu }=0$ in one inertial frame, its Lorentz transformation is also zero in every other inertial frame. The same is true of $K^{\mu }{\tilde{F}}_{\mu \nu }$. Since the transformed wave still has $K^{\prime 0}>0$ and $|{\mathbf{k}}^{\prime }|={\omega }^{\prime }/c$, their temporal components implyHere $K^{\mu }$, rather than a vector renormalized to have temporal component one in every frame, is the actual Lorentz four-vector; multiplying a zero contraction by a nonzero normalization does not affect the argument.

Finally, the two scalar perpendicularity statements alone are not equivalent to one tensor contraction. They do not enforce the amplitude relation between $\mathbf{E}$ and $\mathbf{B}$. Conversely $n^{\mu }{F}_{\mu \nu }=0$ alone cannot exclude a longitudinal magnetic field: adding $B_{\parallel }\mathbf{s}$ leaves its cross product unchanged. For a genuine source-free plane electromagnetic wave, the remaining Maxwell equations supply that missing constraint, and both tensor contractions vanish. This qualification and the corrected index placement are necessary for the intended invariance argument.

First lower the index of the four-current with the Minkowski metric: $J_{\beta }=g_{\beta \alpha }{J}^{\alpha }$. The displayed expression is thenThereforeIts one free lower index transforms with the inverse coordinate Jacobian. The contraction with the metric here lowers an index; it does not make the current a scalar because the index $\delta$ remains free.

The parton model treats a fast proton as an ensemble of approximately collinear, nearly free quarks and antiquarks. A hard deep inelastic scattering collision resolves one parton on a time scale short compared with its binding motion. The probabilities are parton distribution functions; the simplest approximation neglects intrinsic transverse momentum, quark masses and the proton mass in the hard collision. Quantum chromodynamics introduces logarithmic violations of Bjorken scaling, so the parton distribution functions depend on a resolution scale as well as a parton momentum fraction.

Write the incident electron and proton four-momenta as $k,P$, the scattered electron four-momentum as $k^{\prime }$, and set $q=k-k^{\prime }$. With the $(+---)$ Minkowski metric, the Bjorken scaling variable and the inelasticity areAn incident parton with four-momentum $\xi P$ emerges on shell, giving $(\xi P+q{)}^2\simeq 2\xi P\cdot q-Q^2=0$. Thus $\xi =x_B$: the observable Bjorken scaling variable measures the struck parton momentum fraction in this approximation.

For clarity, measure the scattering angle $\vartheta$ from the incoming electron direction, and denote the outgoing electron energy by $E_e^{\prime }$. Direct four-momentum products giveThe expression uses the known beam energies and two measured quantities. An electromagnetic calorimeter measures $E_e^{\prime }$; a tracking detector, the collision interaction vertex and the position of the electromagnetic shower determine $\vartheta$. Coverage close to the incoming electron direction matters at small $Q^2$. A hadronic calorimeter supplies an independent consistency check and rejects backgrounds. If the angle is instead measured from the incoming proton, its sine and cosine half-angle factors interchange.

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The cover establishes Trinity Term 2019 despite the annual bundle's 2018 filename. Use $\hslash =c=1$ and the mostly-minus Minkowski metric. All six questions are solved, including the alternatives allowed by the paper's choice of four questions.

Use the Minkowski metric $(+,-,-,-)$ and choose the incoming electron along $+z$, with the scattered electron in the $xz$ plane. In natural units, neglecting the electron mass, the four-momenta areThe momentum transfer entering the proton is thereforeThe outgoing hadronic system carries $p_X=p_2+q$. The energy transfer is $\nu =E-E^{\prime }$, while $\mathbf{q}$ need not point along the original beam.

An incoming fermion contributes a Dirac spinor $u_s(p)$ and an outgoing fermion contributes its Dirac adjoint ${\bar{u}}_s(p)=u_s^{†}(p){\gamma }^0$. The label $s$ specifies a spin state; $p$ is the on-shell four-momentum. Incoming and outgoing antifermions instead contribute ${\bar{v}}_s(p)$ and $v_s(p)$.

The exchanged boson is a virtual photon, represented in the Feynman rules by the photon propagator $-i{g}_{\mu \nu }/(q^2+iϵ)$ in Feynman gauge. Its momentum transfer is $q=p_1-p_3=p_4-p_2$; unlike an external real photon, it need not satisfy $q^2=0$. Each Feynman vertex contracts a gamma matrix with this propagator and contributes $-ie{\gamma }^{\mu }$ in a conventional QED sign convention. The coupling constant $e$ sets the interaction strength, with fine-structure constant $\alpha =e^2/(4\pi )$. One may reverse the sign of the photon field and both vertex signs without changing the result. The gamma matrices obey $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }I$, with Minkowski metric $g=\mathrm{diag}(1,-1,-1,-1)$.

Matching spacetime symmetries to dynamical symmetries is a powerful demand for a coherent interpretation, but an unconditional equality theorem does not follow from logic or from general covariance. A spacetime symmetry preserves the structure declared physically significant: for Minkowski spacetime, the metric tensor and relevant orientations. A dynamical symmetry takes solutions to solutions with the laws' fixed backgrounds held fixed. The appropriate comparison concerns external transformations on the same physical quantities; internal gauge symmetries have no reason to be isometries of spacetime.

One direction becomes a theorem after important premises are supplied. Suppose all fixed background structure is geometrical and the matter equations are constructed covariantly from it. An automorphism of that full structure then takes solutions of those equations to solutions. For example, the Klein-Gordon equation $({\eta }^{ab}{\partial }_a{\partial }_b+m^2)\varphi =0$, in units $c=\hslash =1$, has the Poincaré symmetry of the fixed Minkowski metric. This is an explanation of symmetry inheritance conditional on a coupling prescription; it does not derive that prescription from the bare existence of a metric tensor.

A counterexample exposes the missing premise. On the same Minkowski spacetime, add a fixed timelike vector field $n^a$ and a matter termFor generic nonzero $\alpha$, a boost which preserves $\eta$ but changes $n$ is not a symmetry with that background held fixed. The full background has fewer symmetries than the metric tensor alone. Writing the equations in arbitrary coordinates and transforming $n$ as a vector field gives general covariance, not Lorentz invariance. Conversely, redundant unobservable geometrical structure could make the declared spacetime symmetry group artificially too small. Accidental symmetries of a restricted matter sector could make its dynamical symmetry group too large.

There are deeper motivations for avoiding such mismatches. If all measuring devices obey Lorentz-invariant dynamics, their behaviour supports a common Minkowski metric interpretation; adding an undetectable preferred rest structure sacrifices explanatory economy. If a preferred structure affects some device, then the original metric-only geometry was incomplete as a description of physical background structure. These observations support either revising the geometry or explaining the universal matter symmetry, but they are methodological and interpretive arguments, not an automatic metaphysical identity.^[ref]

Two qualifications matter. Boundary conditions and initial conditions can select a rest frame in an individual solution while the laws remain boost symmetric. Also, a symmetry of the complete theory need not be detectable by transforming only a subsystem relative to its environment. The best principle therefore matches the physically relevant geometry to the external symmetries of the complete laws, after all backgrounds and couplings are included. Calling it merely a bookkeeping preference understates its explanatory value; calling it an unavoidable consequence of covariance overstates its logical force.

Spacetime geometry gives an acceptable structural explanation of length contraction, provided the relation between material rods, measurement and the metric tensor is included. It is not a force that squeezes a rod, and the mere existence of a quadratic form cannot guarantee how arbitrary matter behaves. The explanatory issue is whether a common geometrical constraint is sufficient for the phenomenon at issue, or whether a constructive theory of the rod is also required.

For an equilibrium rod with proper length $L_0$, use its rest frame $S^{\prime }$, where its endpoint histories are $x_A^{\prime }=0$ and $x_B^{\prime }=L_0$. In a laboratory frame useA laboratory measurement compares endpoints at the same laboratory time. Subtract the two endpoint equations at $\Delta t=0$ to obtainThe two frames measure different pairs of spacetime events. There is no contradiction in each assigning the other's moving rods smaller lengths; their simultaneity relations differ. This is neither an optical distortion nor an invariant compression diagnosed by an attached strain gauge. A photograph also involves unequal emission times and propagation of light, so it is not this simultaneous endpoint measurement.

The geometrical explanation has genuine unifying content. Once all suitable equilibrium rods measure the spatial distances induced by the same Minkowski metric, the result is independent of chemical composition. It relates length contraction to time dilation and relativity of simultaneity, rather than postulating separate mechanisms for every device. A structural explanation need not be a temporal causal story: the result compares descriptions of an existing world tube, not successive states of a rod being accelerated.

The main objection asks what warrants the italicized measurement premise. Matter on a mathematical Minkowski spacetime could obey equations involving an additional preferred vector field, as in 3. Such matter need not form devices with the usual transformation behaviour. Harvey R. Brown and Oliver Pooley argue that a constructive understanding must attend to the Lorentz-invariant dynamics of matter.^[ref] This objection has force against treating a bare metric tensor as an independently acting agent. It does not invalidate the endpoint derivation once the physical metric interpretation is established.

Acceleration introduces a different explanatory task. A rod pushed at one end can undergo stresses, heating and oscillations; its final proper length can change. A dynamical calculation is needed to determine whether it settles into an equivalent equilibrium state. The comparison $L=L_0/\gamma$ assumes a specified equilibrium proper length, and does not assert that every acceleration history preserves it. Geometry constrains the measurement of that state; material dynamics explain its formation and stability. Those explanatory roles can coexist without making either empty.

An observer-independent global presentism conflicts with the absence of a distinguished simultaneity relation in bare special relativity. It is not a formal contradiction to add a preferred foliation. The issue is what the proposed present contains, and whether that extra structure is part of the physical theory or an additional metaphysical commitment. Presentism must be distinguished from the harmless indexical fact that an observer calls their own current event “now”.

Let two spacetime events have coordinate differences $\Delta t=0$, $\Delta x=L\ne 0$ in an inertial frame. A relatively moving frame assignsThey are simultaneous in one frame and not in another. For a fixed event $P$, each inertial four-velocity $u$ defines a different simultaneous hyperplane through $P$ by $\eta (u,Q-P)=0$. No particular $u$ is selected by the Minkowski metric alone. Thus an extended objective present cannot both be the equal-time slice of one ordinary inertial frame and be invariant under all Lorentz transformations.

An equivocation can make the incompatibility seem stronger than it is. If one defines “real together” using one observer's simultaneity, then changes observers in the next inference, transitivity across the mixed relations has not been justified. It is the failure of transitivity for mixed-frame simultaneity. The physics supplies the frame-indexed relations; it does not supply a single absolute co-existence relation formed by silently joining all of them. A sound argument against relativistic presentism challenges the physical basis of its preferred relation, rather than proving all events real by that equivocation.

There are several coherent responses, with different costs. A preferred foliation can be appended to Minkowski spacetime, declaring one of its slices the objectively existing present. If it affects no physical process, it has no support from the relativistic laws and breaks the symmetry of the total ontology without changing observable predictions. If it affects matter, new laws must explain why ordinary tests display Lorentz invariance. This is additional structure, not a logical impossibility. Eternalism, by contrast, treats the complete spacetime history as real and makes simultaneity relational, but the relativity of coordinate time by itself does not deductively establish that metaphysics.

A local present containing only $P$ is invariant once $P$ is given, yet it abandons a spatially extended world existing now. The past light cone is invariant too, but contains events of different ages and is not an instantaneous spatial present. An observer-relative present is meaningful, but relativizes the existence claim rather than defending a unique global present. The defensible conclusion is therefore an incompatibility among bare relativistic structure, an objective extended present, and full symmetry of existence claims; it is not an unconditional proof that every formulation of presentism is incoherent.

The numerical coordinates assigned to an event can be changed without changing the event. It does not follow that every relation between coordinate systems is devoid of physical content. A standard inertial frame includes a family of resting measuring standards and synchronized clocks. Its coordinates are normalized so that the Minkowski metric has the same standard form. The relation between two such descriptions therefore records properties of the standards and of spacetime, not just an arbitrary choice of names.

Write $x^{\mu }=(ct,x,y,z)$ and use metric signature $(-,+,+,+)$. A Lorentz transformation satisfiesAn origin translation can also be included without changing the Minkowski interval. The displayed condition preserves the interval between events, including the null cone and proper time along a timelike curve. Its physical significance lies in the claim that comparisons made with actual idealized rods and clocks obey this same metric structure.

A more general passive relabeling is allowed, but then the metric components must change with the labels. For example, let $t^{\prime }=2t$ and leave $x,y,z$ fixed. In the new coordinates,Nothing physical has changed, but $t^{\prime }$ no longer has the original clock-normalized meaning of $t$. Retaining the old metric components as well as the new labels would change the mathematical model. This distinction is part of the physical content of a passive Lorentz transformation: the transformation between standard inertial coordinate systems preserves a specified metric form, whereas arbitrary relabelings need not preserve that form.

For an operational example, a clock moving at velocity $v$ between two events on its world line accumulatesin the first frame. In its own rest frame the same result is simply the interval between its local readings. The equality is a physical agreement about one clock, derived using the preserved Minkowski interval. It is not a claim that the two coordinate-time differences are identical. Similarly, an observer measures a light angular frequency through $\omega =-k_{\mu }{u}^{\mu }$, where $k^{\mu }$ is the wave four-vector and $u^{\mu }$ the observer's four-velocity. Transforming both sets of components leaves this scalar unchanged. Comparing two different observers gives the physical relativistic Doppler effect rather than a disagreement over one observer's measurement.

There is a further distinction between passive transformations of the same situation and active changes to a situation. Re-expressing a particular experiment in different coordinates is always possible if all geometric fields are transformed consistently. Preparing a second, moving copy of the apparatus is a different physical arrangement. Its relation to the first is governed by a dynamical symmetry only if the relevant laws and background structure have the appropriate invariance. An active Lorentz transformation of a vector field makes the changed field configuration explicit.

This is why mere Lorentz covariance of notation must not be confused with the Lorentz invariance of the dynamics. A physically preferred timelike vector field can be transformed as a vector under passive relabeling while distinguishing particular states of motion. Holding that preferred field fixed can destroy the active boost symmetry of the matter dynamics. The covariant equation with a preferred timelike vector is a concrete example.

Coordinates are therefore labels, but the Lorentz transformation relates labels tied to shared physical standards and a definite metric structure. Its measurable consequences follow from that structure and the dynamical assumptions, not from the bare act of changing labels.^[ref]

The crucial distinction is between an observer-independent binary relation of absolute simultaneity and a relation defined relative to an inertial frame. The relativity of simultaneity removes the former from the usual structure of special relativity; it does not make the latter meaningless or subjective.

For two spacetime events separated by $(\Delta t,\Delta x)$, a Lorentz boost of velocity $v$ givesIf $\Delta t=0$ and $\Delta x\ne 0$, then $\Delta t^{\prime }\ne 0$ for a nonzero boost along their separation. More generally, spacelike separated events can be simultaneous in some inertial frame, and their time order can reverse between inertial frames. Distinct timelike separated or null separated events cannot be simultaneous in any inertial frame; their causal order is preserved by time-orientation-preserving Lorentz transformations. These are precise geometric facts, independent of anyone's beliefs.

In Minkowski simultaneity relative to an inertial observer, the observer is represented by a constant future-directed four-velocity $u$, normalized by $\eta (u,u)=-c^2$ in metric signature $(-,+,+,+)$. ThenFor a fixed $u$, this is a perfectly well-defined equivalence relation. It can be specified without coordinates. What the Minkowski metric does not supply is a uniquely preferred $u$ and hence a uniquely preferred simultaneity foliation. This is the strongest defensible interpretation of the claim by Tim Maudlin: there is no additional fundamental, frame-independent simultaneity relation in the bare theory.

Calling it merely “relative to an observer” can mislead in two ways. First, distant simultaneity does not depend on when an image is received: travel time of the light signal must be accounted for. Second, two observers at the same spacetime event but with different four-velocities assign different simultaneity hyperplanes; the dependence is on state of motion, rather than consciousness. Einstein synchronisation operationalizes the standard frame assignment by setting a reflected pulse's remote reading to the midpoint of its sending and returning readings.^[ref]

A material system can pick out a useful rest frame, for example a gas in thermal equilibrium. That adds physical information beyond the Minkowski metric; it does not provide a universal absolute present. Accordingly, Maudlin's denial is justified if it denies a primitive observer-independent relation, but is too strong if read as denying every objective relation $P{\sim }_{u}Q$. The indexed relation and its measurable consequences remain part of the physics.

There are two distinct explanatory questions. One asks why a proposed system of equations has a particular dynamical symmetry. The other asks why the common spacetime geometry is the right geometry for the behavior of physical rods, clocks and fields. A geometric account and a dynamical account give different directions of explanation, and their merits cannot be decided by a change of notation alone.

For the geometric account, Minkowski spacetime has a fixed Lorentzian metric; its spacetime isometries form the Poincaré group. If the matter equations use that metric tensor and no additional symmetry-breaking background structure, with couplings and boundary assumptions appropriate to the transformation, these spacetime isometries take solutions into solutions. For example, the Klein-Gordon equationin metric signature $(-,+,+,+)$ is invariant under the Poincaré group. This provides one common reason for the same transformation rules appearing in many laws. An individual solution, such as a particular plane wave, need not possess every symmetry of the equation.

Geometry alone is insufficient unless the assumed coupling to geometry is included. In the covariant equation with a preferred timelike vector,the fixed timelike $n$ selects a preferred rest frame. All the quantities can be transformed consistently under a coordinate transformation, but boosts that change $n$ are not symmetries with the physical background fixed. Thus general covariance, tensor notation and the mere presence of the Minkowski metric do not by themselves establish Einstein's principle of relativity.

A dynamical account instead starts with the common invariance of the actual interactions and asks why their composite rods and clocks measure a common interval. If their equations and suitable equilibrium solutions transform consistently under Lorentz transformations, their moving configurations can exhibit length contraction and their periodic processes time dilation. This is the explanatory strategy defended by Harvey R. Brown and Oliver Pooley, who deny that merely redescribing these effects geometrically supplies their microscopic explanation.^[ref] Such an account must still explain why independent interactions have the same relevant symmetry and why a particular clock is a good clock; it cannot simply infer those facts from the word “dynamics”.

The positions therefore need not disagree about experimental predictions. The geometric presentation makes a shared constraint explicit and permits economical deductions. The constructive presentation investigates why material systems realize that constraint. Defining geometry from universal dynamical behavior is a substantive proposal; deriving behavior from geometry also requires substantive matter-coupling assumptions. Neither direction is made compulsory by the formal equivalence of the descriptions.

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Solutions use natural units $\hslash =c=1$ and Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$. The d'Alembert operator is $\square ={\partial }_{\mu }{\partial }^{\mu }$. All four questions are covered, including the optional fourth question.

For incoming four-momenta, the momentum-space Feynman rules in the chosen Minkowski metric convention are:

There is one connected one-loop one-particle irreducible two-point graph: a single quartic vertex with two external lines and one tadpole diagram. Its symmetry factor is $S=2$ from interchange of the two loop half-edges.

There are three connected one-loop one-particle irreducible four-point graphs: two quartic vertices joined by two parallel internal lines, with external partitions $(12|34)$, $(13|24)$ and $(14|23)$. These are the $s$, $t$ and $u$ scattering channels; each bubble diagram has symmetry factor $S=2$. External-branch insertions are excluded by the one-particle irreducible requirement.

The positive denominators in this part are Euclidean. Starting with the Minkowski spacetime Feynman rules, first perform a Wick rotation; a bare Minkowski metric integral without a pole prescription would not equal the displayed positive expression. Work initially with $m^2>0$ and $\Re n>D/2$, where the Euclidean massive loop integral converges.

Use Schwinger parameterization followed by the Gaussian integral:The remaining Gamma function integral givesThe factorial form is for positive integral $n$. Dimensional regularization defines the result elsewhere by analytic continuation.

For the scalar bubble integral, combine the two denominators with a Feynman parameter $x$ and shift the integration variable to $\ell =k-(1-x)p_E$. ThenFor $D=4-ϵ$ this isThe Gamma function recurrence gives $\Gamma (ϵ/2)=2/ϵ+O(1)$, and the parameter integral is $1+O(ϵ)$. ConsequentlyThis ultraviolet divergence is independent of the external momentum. The assumption $m^2>0$ avoids an infrared divergence. In particular the completely massless zero-momentum integral is a scaleless integral and vanishes in dimensional regularization; its ultraviolet and infrared poles must be separated before assigning it the displayed ultraviolet residue.

Source paper: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/5/Physics%20Papers%202016.pdf?cookie_passthrough=1. The cover identifies Hilary Term 2017, 11 January; the bundle's 2016 filename does not determine the examination year. We use natural units $\hslash =c=1$, the mostly-minus Minkowski metric, and Feynman propagators with the $+i0$ prescription.

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The scanned cover establishes Hilary Term 2018, despite the containing bundle's 2017 filename. Use $\hslash =c=1$, the Minkowski metric signature $(+,-,\dots ,-)$ outside question 2, which uses Euclidean space, and $p\cdot x=E_{\mathbf{p}}t-\mathbf{p}\cdot \mathbf{x}$. The phase and normalization conflicts in question 4 are identified in a; its path integral is evaluated with the normalization of its displayed action.

Source paper: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/7/Physics%20Papers%202018.pdf?cookie_passthrough=1

The cover establishes Hilary Term 2019 despite the bundle's 2018 filename. Use $\hslash =c=1$. For questions 2 and 3 the Minkowski metric has signature $(+,-,\dots ,-)$. Question 1 has a preferred time coordinate and does not assume Lorentz invariance.

A continuous Lorentz transformation in one spatial dimension must preserve $t^2-x^2$. In the component connected to the identity, take the image of the time axis to have velocity $v$, with $|v|<1$, and write $B_v(1,0{)}^{\mathsf{T}}=\gamma (1,v{)}^{\mathsf{T}}$. Preservation of the Minkowski norm gives ${\gamma }^2(1-v^2)=1$, so $\gamma =(1-v^2{)}^{-1/2}$. The spacelike column must be orthogonal to the timelike column and have squared Minkowski norm $-1$; continuity fixes it to $\gamma (v,1{)}^{\mathsf{T}}$. ThusHere $x=(t,x{)}^{\mathsf{T}}$ denotes the coordinate pair. This is the positive off-diagonal convention in the source; describing a passive frame moving at $+v$ instead reverses the sign of $v$.

Define the rapidity $\alpha =\mathrm{artanh}v$. The hyperbolic function identities giveIn light-cone coordinates, $t^{\prime }+x^{\prime }=e^{\alpha }(t+x)$ and $t^{\prime }-x^{\prime }=e^{-\alpha }(t-x)$. Therefore the derivatives transform oppositely:Use the field-transformation convention ${\psi }^{\prime }(x^{\prime })=S(\alpha )\psi (x)$. The component action in c is invariant underIndeed, the right-component kinetic term gains $e^{\alpha /2}{e}^{-\alpha }{e}^{\alpha /2}=1$, and the left-component kinetic term gains $e^{-\alpha /2}{e}^{\alpha }{e}^{-\alpha /2}=1$. The two factors in each Dirac mass term cancel as well. Finally $\det B=1$, so $d{t}^{\prime }d{x}^{\prime }=dtdx$ and the whole action is unchanged. These are the chiral boost weights in two spacetime dimensions.

In spinor notation, $S(\alpha )=\exp (\alpha {\gamma }^5/2)$ satisfies $S^{-1}{\gamma }^{\mu }S=B^{\mu }{}_{\nu }{\gamma }^{\nu }$ and $S^{†}{\gamma }^{0}S={\gamma }^0$, so ${\bar{\psi }}^{\prime }=\bar{\psi }{S}^{-1}$. These identities directly verify covariance of the Dirac equation and the Lorentz invariance of its action. The matrices $S$ are not unitary with respect to the ordinary component norm; the Dirac adjoint supplies the invariant bilinear.

The statement about there being only Lorentz boosts applies to the identity component of the One-dimensional Lorentz group. The full Lorentz group also has discrete transformations: the concrete spatial reflection $\mathrm{diag}(1,-1)$ preserves the Minkowski metric but has determinant $-1$, so it cannot equal any displayed $B(\alpha )$, whose determinant is $+1$.