Relativistic Uncertainty Principle

[LarryS]

In non-relativistic wave mechanics, the momentum-position uncertainty relationship and the energy-time relationship exist because these variables are related via the Fourier Transform of the wave function.

Is there a relativistic (QFT) equivalent or analog of the above px and Et uncertainty relationships?

As always, thanks in advance.

 

[Fredrik]

Everything in this post is valid in both relativistic and non-relativistic QM. And the stuff mentioned in George Jones's post here is too. (The stuff in Galileo's post definitely is. I haven't looked closely at what Messiah is saying, but it appears to be valid too).

There is however an issue with the existence of a position operator in relativistic QM. Look up the "Newton-Wigner position operator" if you're interested.

 

[LarryS]

Everything in this post is valid in both relativistic and non-relativistic QM. And the stuff mentioned in George Jones's post here is too. (The stuff in Galileo's post definitely is. I haven't looked closely at what Messiah is saying, but it appears to be valid too).

There is however an issue with the existence of a position operator in relativistic QM. Look up the "Newton-Wigner position operator" if you're interested.

That was very informative. But is it possible to represent both the momentum-position and Energy-time uncertainties in one inequality referencing the 4-position and 4-momentum vectors of SR?

 

[Demystifier]

That was very informative. But is it possible to represent both the momentum-position and Energy-time uncertainties in one inequality referencing the 4-position and 4-momentum vectors of SR?

Yes, provided that you enlarge the Hilbert space such that a state in the Hilbert space is a function not only of space but of both space and time. See
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595]

 

[Fredrik]

There's no time operator and therefore no 4-position operator in standard QM. For massless particles there isn't even a 3-position operator. Also, the position operator that exists for massive particles (Newton-Wigner) is frame dependent: If the particle is localized in one inertial frame, it's not in others.

 

[Demystifier]

There's no time operator and therefore no 4-position operator in standard QM. For massless particles there isn't even a 3-position operator. Also, the position operator that exists for massive particles (Newton-Wigner) is frame dependent: If the particle is localized in one inertial frame, it's not in others.

That's true in the sense that you cannot construct the corresponding operators in terms of physical states, where "physical" means solutions of the corresponding wave equations (Schrödinger, Klein-Gordon, etc.) of motion. This means that DYNAMICAL operators don't exist. Nevertheless, KINEMATIC operators (constructed from mathematical wave functions that do not necessarily satisfy the wave equations of motion) exist. Another useful terminology is that on-shell operators do not exist, but off-shell operators exist.

Let me also briefly describe how these operators could in principle be even on-shell. Assume that we have a slightly more general physical theory, in which the mass squared is not a fixed parameter, but an operator with both positive and negative eigenvalues. For some reason the states that we currently observe are eigenstates of this operator with only non-negative values, but negative values are also possible in principle. The most general state is a superposition with both positive and negative mass-squared states. In such a hypothetic physical theory, both time and space position operators would be physical.