QuantumPhysics.dvi

spin waves, phonons, spinons etc). Here, however, we shall concentrate onrelativistic quantum field
theorybecause relativity forces the number of particles not to be conserved. In addition, relativity
is one of the great fundamental principles of Nature, so thatits incorporation into the theory is
mandated at a fundamental level.

18.3 Further conceptual changes required by relativity

Relativity introduces some further fundamental changes inboth the nature and the formalism of
quantum mechanics. We shall just mention a few.

• Space versus time
  In non-relativistic quantum mechanics, the positionx, the momentumpand the energyEof
  free or interacting particles are all observables. This means that each of these quantitiesseparately
  can bemeasuredto arbitrary precision in an arbitrarily short time. By contrast, the accurarcy of
  the simultaneous measurement ofxandpis limited by the Heisenberg uncertainty relations,

∆x∆p ∼ ̄h

There is also an energy-time uncertainty relation ∆E ∆t ∼ ̄h, but its interpretation is quite
different from the relation ∆x∆p ∼ ̄h, because in ordinary quantum mechanics, time is viewed
as a parameter and not as an observable. Instead the energy-time uncertainty relation governs
the time evolution of an interacting system. In relativistic dynamics, particle-antiparticle pairs
can always be created, which subjects an interacting particle always to a cloud of pairs, and thus
inherently to an uncertainty as to which particle one is describing. Therefore, the momentum itself
is no longer an instantaneous observable, but will be subject to the a momentum-time uncertainty
relation ∆p∆t∼ ̄h/c. Asc→ ∞, this effect would disappear, but it is relevant for relativistic
processes. Thus, momentum can only be observed with precision away from the interaction region.

Special relativity puts space and time on the same footing, so we have the choice of either
treating space and time both as observables (a bad idea, evenin quantum mechanics) or to treat
them both as parameters, which is how QFT will be formulated.

• “Negative energy” solutions and anti-particles
  The kinetic law for a relativistic particle of massmis

E^2 =m^2 c^4 +p^2 c^2

Positive and negative square roots forEnaturally arise. Classically of course one may just keep
positive energy particles. Quantum mechanically, interactions induce transitions to negative energy
states, which therefore cannot be excluded arbitrarily. Following Feynman, the correct interpreta-
tion is that these solutions correspond to negative frequencies, which describe physical anti-particles
with positive energy traveling “backward in time”.

• Local Fields and Local Interactions
  Instantaneous forces acting at a distance, such as appear inNewton’s gravitational force and
  Coulomb’s electrostatic force, are incompatible with special relativity. No signal can travel faster