The violation of energy conservation law?

[ndung200790]

Please teach me about this question:
In a process of scattering of closed system of particles(in point of view of quantum field theory),the Lagrangian of the system is not invarian(up to 4-divergence) under time translation.So that following the Noether theorem,the energy of the system is not conservation.I think this is a paradox.
Thank you very much for advanced.

 

[Demystifier]

... the Lagrangian of the system is not invarian(up to 4-divergence) under time translation.

What makes you think so?

 

[ndung200790]

I think that before collision the Lagrangian is the noninteraction Lagrangian,but in the process of scattering the Lagrangian is the interaction Lagrangian.So that may be the time translation would not be symmetry translation.Please be patient teach me again.
Thank you very much.

 

[fermi]

I think that before collision the Lagrangian is the noninteraction Lagrangian,but in the process of scattering the Lagrangian is the interaction Lagrangian.So that may be the time translation would not be symmetry translation.Please be patient teach me again.
Thank you very much.

I have two comments for you:

1) The Lagrangian before, during, and after the collision is not different. It is the same Lagrangian in all cases, which includes free and interaction terms. It so happens that the interaction terms are usually very very small when the particles are far apart from each other. (Excluding the example of two free quarks of course, but confinement takes care of that.)

2) The free Lagrangian as well as the interaction Lagrangian are both invariant under time translations. Your statement that they are not invariant is just not true for any field theory that I know of. In fact they are not only invariant under time translations, but they are also invariant under space translations as well. We require all field theories to be Poincare Group invariant, which includes all Lorentz transformations as well as all space-time translations. You could in principle write down a Lagrangian that is not Poincare invariant. No one to my knowledge have found any evidence for such a Lagrangian describing any collision process that we have observed.

 

[dx]

Scattering processes in quantum mechanics conserve energy only when the time of the scattering is completely uncertain. Or, more precisely, phenomena which are analyzed in terms of energy conservation cannot appear in situations where the time of the scattering is fixed. To illustrate, let a fixed heavy object at x = 0 scatter a particle of momentum k₁ to a particle of momentum k₂. The amplitude for this scattering is (g small, and ignoring higher order terms)

$$g\langle k_2| \sum_t \Psi^{\dagger}_{t}(0)\Psi_{t} (0)| k_1 \rangle = g \sum_t e^{i(\omega_2 - \omega_1)t} = g2\pi \delta(\omega_2 - \omega_1)$$

Thus, the amplitude is finite only if $\omega_1 = \omega_2$, i.e. energy is conserved.

 

[piareround]

As some of the people in the particle physics thread can explain better than, I there are a few particle decay reactions that violate energy conservation such as the p + e --> n + v_e reaction(*). However, all these reactions have a very specific restriction in that they cannot occur out side the size of the nucleous itself.
Yes, these reactions occur thanks to quantum leaps in energy and uncertainty; however, due to a correlations between energy an time we correlate the bond energy or center mass energies (**) measured to a specific reaction decay time or half-life of the particle. *At least I think its http://en.wikipedia.org/wiki/Electron_capture"
**Sorry I am learning to particle physics myself show I cannot rememeber which one it is >>;