Probability conservation and symmetry

In summary, the theorem of Wigner states that any symmetry of a physical system will ensure conservation of probabilities. This is because symmetries are defined by the fact that there is no observational difference between the system and the symmetry applied to it. This means that probabilities must remain the same before and after the symmetry is applied. Mathematically, this translates to symmetries being represented by unitary or anti-unitary operators in rigged Hilbert space language. This is why symmetries are crucial in determining the quantum description of a system.
  • #1
domhal
4
0
What symmetry gives probability conservation? Or, what symmetry does probability conservation give?

I have been trying (unsuccessfully) to find an answer to this question. I think the question makes sense. That is, I can't see how the situation is different from, for example, that of spatial translation giving conservation of momentum.
 
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  • #2
Unfortunately I don't have time to flip out a complete answer, but I would tell you to look at the Lagrangian formulation of the Schrodinger equation and to look up Noether's theorem. I seem to recall actually working this out at some point, but those notes are probably buried in a box somewhere.
 
  • #3
Actually the theorem of Wigner insures probability conservation for any symmetry of the system. By that theorem, symmetry transformations are implemented in (rigged) Hilbert space language by either unitary or antiunitary operators which are known to conserve scalar products, hence probabilities.
 
  • #4
I can't see how the situation is different from, for example, that of spatial translation giving conservation of momentum.
I think probablity conservation looks fundamentaly different from mometum conservation. I recall momentum conservation being derived by assuming that Hamilton's operator commutes with a translation operator. For one particle state, [tex][e^{u\cdot\nabla},H]=0[/tex] for abritary vector u. From this it follows, that [tex][-i\hbar\nabla,H]=0[/tex], and with Shrodinger's equation, that [tex]\langle\Psi|-i\hbar\nabla|\Psi\rangle[/tex] is conserved in time. Analogously to this, you could argue that quantity [tex]\langle\Psi|\Psi\rangle[/tex] is conserved simply because [tex][1,H]=0[/tex], but that looks dumb. These don't look the same kind of conservation laws.
 
  • #5
jostpuur said:
I think probablity conservation looks fundamentaly different from mometum conservation. I recall momentum conservation being derived by assuming that Hamilton's operator commutes with a translation operator. For one particle state, [tex][e^{u\cdot\nabla},H]=0[/tex] for abritary vector u. From this it follows, that [tex][-i\hbar\nabla,H]=0[/tex], and with Shrodinger's equation, that [tex]\langle\Psi|-i\hbar\nabla|\Psi\rangle[/tex] is conserved in time. Analogously to this, you could argue that quantity [tex]\langle\Psi|\Psi\rangle[/tex] is conserved simply because [tex][1,H]=0[/tex], but that looks dumb. These don't look the same kind of conservation laws.

They're actually very similar, in that they are related to Noether's Theorem and "conserved charges". This is why I was saying to look at the lagrangian.
 
  • #6
dextercioby said:
Actually the theorem of Wigner insures probability conservation for any symmetry of the system. By that theorem, symmetry transformations are implemented in (rigged) Hilbert space language by either unitary or antiunitary operators which are known to conserve scalar products, hence probabilities.

From a physical point of view, I would even say: the argument goes in the other way! Because the physical manifestation of a symmetry is, well, that there is no observational difference between the system as such, and the "symmetry candidate" applied to the system, and because all observables are essentially expectation values, which are essentially weighted probabilities, the necessary and sufficient condition for a symmetry candidate on a physical system to be actually a symmetry, is that all probabilities "before" and "after" are the same. Hence, symmetries NEED to conserve probabilities (and hence all observable phenomena) in order for them to merit the denomination of symmetry.

And from this condition follows mathematically that they must be unitary or anti-unitary representations of their group (because symmetries also always form a group, for the same physical reasons).

EDIT: we can, because of this, apply two kinds of approaches to setting up the quantum description of a system. We can make a list of symmetries, and try to find a good representation of them, which can then serve as a quantum description ; or we can have another way of finding the quantum description, and then go fishing for the different symmetry representations it contains.
 
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  • #7
Yes, Vanesch, you're right, that's why the original question

"What symmetry gives probability conservation? Or, what symmetry does probability conservation give?"

should be answered: "Any symmetry. All symmetries".
 

1. What is probability conservation?

Probability conservation is the principle that states that the total probability of a system remains constant over time. This means that the sum of all possible outcomes for a given event must always equal 1.

2. How does probability conservation relate to symmetry?

Probability conservation is closely linked to the concept of symmetry. Symmetry refers to the invariance of a system under certain transformations, such as rotation or reflection. If a system is symmetrical, then the probability of an event occurring in one state should be the same as the probability of the event occurring in a symmetrical state. This is known as symmetry conservation.

3. Why is probability conservation important in science?

Probability conservation is an important concept in science because it allows us to make predictions about the behavior of systems. By understanding the symmetry and conservation laws of a system, we can make accurate predictions about the likelihood of certain events occurring.

4. How is probability conservation applied in different fields of science?

Probability conservation is applied in a variety of fields, including physics, chemistry, and biology. In physics, it is used to describe the behavior of particles and energy in various systems. In chemistry, it helps to explain the stability and reactivity of molecules. In biology, it is used to model and predict the behavior of populations and ecosystems.

5. Are there any exceptions to probability conservation?

While probability conservation is a fundamental principle in science, there are certain situations where it may not hold true. For example, at the quantum level, particles can have a non-zero probability of spontaneously appearing or disappearing, violating the conservation of energy and momentum. However, these events are still governed by the laws of probability conservation on a larger scale.

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