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metroplex021
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Hi folks -- does anyone know of a proof that particle (quanta) number in QFT is / is not Lorentz invariant? I'd be happy to hear of it -- so thanks!
Bill_K said:Particle number is NOT Lorentz invariant. Particle number is defined with respect to a particular spacelike hypersurface, and specifying its value includes specifying the hypersurface. Particle number at zero time in one reference frame does not necessarily equal the particle number at zero time in another reference frame.
metroplex021 said:Hi folks -- does anyone know of a proof that particle (quanta) number in QFT is / is not Lorentz invariant? I'd be happy to hear of it -- so thanks!
The particle number is not a constant in a QFT with interactions. Such as φ4 theory.samalkhaiat said:The number operator is a generator of [itex]U(1)[/itex] symmetry, like the electric charge, the baryon number and the lepton number. These are Lorentz invariant scalars and one can prove this.metroplex021 said:Hi folks -- does anyone know of a proof that particle (quanta) number in QFT is / is not Lorentz invariant? I'd be happy to hear of it -- so thanks!
Bill_K said:The particle number is not a constant in a QFT with interactions. Such as φ4 theory.
You're confusing time translation with time evolution. The way to get the state at a later time t is to use the evolution operator, U(t) = exp(iHt), where H is the Hamiltonian. And in the general case, H contains terms like φ4 which don't conserve particle number.Einj said:Could you explain that better? As far as I know the action of the representation of the Lorentz group, [itex]U(\Lambda)[/itex], on a N-particle state is:
$$
U(\Lambda)|p_1,p_2,\cdots,p_N\rangle=|\Lambda p_1,\Lambda p_2,\cdots,\Lambda p_N\rangle.
$$
I don't see how this can create/destroy particles. What am I doing wrong?
Vanadium 50 said:The confusion is caused by the ill-defined "particle number". If you define particle number as the number of particles in a box, this is invariant, although the shape of the box is not. If you define particle number as the number of particles at some time t, it is not invariant.
No, I'm saying they're two different things. Lorentz transformations are coordinate changes. Time translation means you replace t by t + t0. A system is time-translation invariant if it does not depend explicitly on the value of t. You pick the system up and put it down unchanged. The switch from standard time to daylight sayings time is a time translation.Einj said:From what I understand you are saying that we also must consider the evolution operator as the action of a Lorentz transformation. Is that right?
Bill_K said:The particle number is not a constant in a QFT with interactions. Such as φ4 theory.
In an interacting theory (constructed according to the instant form of dynamics), the Lorentz boost operator contains another term determined by the interaction (just as the Hamiltonian has another term compared to the free theory).Einj said:Shouldn't all the possible Lorentz transformations be represented by the representation U?
Yeah -- he "hides" it under "Symmetries of the S-Matrix", section 3.3. (That's where you can find stuff about modifications to the Lorentz boost operator for interacting theories.)metroplex021 said:Strangerep, do you have the section ref for Weinberg? I can't seem to find thie discussion of this.
A particle number Lorentz invariant is a physical quantity that remains constant regardless of changes in the frame of reference. It is a fundamental property of particles in the theory of special relativity.
Particle number Lorentz invariance is important because it allows for a consistent and universal way to describe the behavior of particles in different frames of reference. It also plays a crucial role in the development of quantum field theories.
Particle number Lorentz invariance is closely related to the conservation of particle number, as it ensures that the total number of particles in a system remains constant regardless of the observer's frame of reference.
In certain physical systems, such as those involving strong interactions, the violation of particle number Lorentz invariance has been observed. However, it is still a fundamental principle in most areas of physics and is considered a key aspect of our understanding of the universe.
Yes, there are several experimental tests for particle number Lorentz invariance, including high-energy particle collisions and measurements of particle lifetimes. These tests have shown that particle number Lorentz invariance holds to a high degree of precision in most physical systems.