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Based in part on my reading of the gospel according to Landau, here is a list of preserved/rejected ideas, that do/do not survive the transition from Special Relativity to Quantum Field Theory:

  • preserved: Lorentz-invariance
  • preserved: energy, momentum, angular momentum of a free particle (can be measured with arbitrary accuracy (given enough time)
  • preserved: c is a speed limit
  • rejected: the path of an object; can't be used in QM, since the idea of a path is rejected.
  • rejected: q, the coordinate of a thing; in QFT, it can't be used as the independent variable of a wave function. (This comes from the uncertainty relations, combined with finite c, leading to a lower limit on the precision with which q can be measured.)

Q1: how would you correct the above?

Q2: event coords, space-time interval: if the coordinate q is rejected, then are these also rejected in QFT?

Q3: I have seen remarks in different places about faster-than-light virtual particles. Does that mean that c-as-a-speed-limit is violated, at some level, in QFT?

John
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1 Answers1

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Q1: the bullets are correct. To add some details,

  • Lorentz covariance, or more properly, PoincarĂ© covariance, is one of the basic pillars of the theory, and one of the main motivations to use fields at all.

  • The existence of energy-momentum and angular momentum is a direct consequence of the PoincarĂ© covariance of the theory (see above). The situation is essentially the same as in standard QM: states with definite momentum, plane waves, are an idealisation that cannot be realised in practice. Plane waves are not normalisable, but we can consider wave-packets, which can be arbitrarily sharp in momentum space.

  • The speed of causality, $c$, is observer-independent, because of PoincarĂ© covariance (see above). Nothing can travel faster than $c$, not even "quantum fluctuations", whatever that means. There is no violation of causality through quantum effects.

  • The path of an object is an ill-defined concept in QM. Add relativity into the mix, and it gets even more complicated.

  • There are no wave-functions in QFT.

Q2: no, space-time coordinates are not rejected. Rather, they are implemented differently than in standard QM. It's a whole new paradigm. In standard QM you have a position operator $\hat X$; if you wanted to extend this into a covariant theory, you'd have to introduce a time operator too, $\hat T$. But such an operator is known to be problematic (cf. this PSE question).

The new paradigm consists on the following: instead of keeping the position operator, $\hat X$, we make the rest of operators position-dependent too. Therefore, a generic operator becomes $\hat A=\hat A(t,\boldsymbol x)$. Thus, the space-time coordinates become labels on operators instead of the eigenvalues of $\hat T,\hat X$.

Q3: no, the speed limit is not violated. Virtual particles are just contractions of fields in the Dyson expansion of the $S$ matrix (cf. this answer of mine). Virtual particles are not particles, they are just a mathematical device that simplifies the formalism. See also Do virtual particles actually physically exist? and What actually are virtual particles?.

In any case, you can rest assured that there is no violation of causality in QFT. Nothing propagates faster than light. Actually, causality is another one of the basic pillars of QFT. In a broad sense, it can be said that QFT is the most general theory that satisfies Poincaré covariance, causality, and the principles of quantum mechanics.

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