Do the degrees of freedom change for particles in a relativistic system?

In summary, when considering one relativistic particle, the configuration space has 3 degrees of freedom due to the mass relation constraint. However, when the particle decays into 2 particles, the configuration space is further reduced due to the conservation of energy and momentum. This leads to a decrease in the degrees of freedom for the system.
  • #1
LarryS
Gold Member
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Consider one particle traveling at a relativistic velocity in 3-space. Then the configuration space of the system consisting of that one particle would have 3 degrees of freedom – 1 particle times 3 dimensions.

Because of its high energy, the particle decays into, say, 2 particles. Now the configuration space of the system has 6 degrees of freedom.

Is the above correct?

But, the 2 particles are now entangled in position (as well as momentum). So does the configuration space truly have 6 degrees of freedom, or something less than that due to position correlation between the 2 particles?

Thanks in advance.
 
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  • #2
referframe said:
Consider one particle traveling at a relativistic velocity in 3-space. Then the configuration space of the system consisting of that one particle would have 3 degrees of freedom – 1 particle times 3 dimensions.
For a relativistic particle, there are two invariants: the mass and the (total) spin. For simplicity, let's suppose we're talking about a spinless particle. Then, although it might seem like there's 4 degrees of freedom (1 energy and 3 momenta), these are constrained by the relativistic mass relation:
[tex]
M^2 ~=~ E^2 - |{\mathbf P}|^2
[/tex]
so there's really only 3 degrees of freedom. The equation above defines the "mass hyperboloid", or "mass shell" for the particle.

Because of its high energy, the particle decays into, say, 2 particles. [...]
Total energy and total momentum are both conserved. This places further constraints on any particles that might emerge from a decay, thus reducing the degrees of freedom applicable to such situations.
 

1. What is the concept of degrees of freedom in quantum mechanics?

Degrees of freedom in quantum mechanics refer to the number of independent variables that can affect the behavior of a quantum system. In other words, it is the number of ways a system can move or change.

2. How are degrees of freedom related to the uncertainty principle?

The uncertainty principle states that it is impossible to know both the position and momentum of a particle simultaneously. This is because the more precisely we know one quantity, the less we know about the other. Degrees of freedom play a role in this principle as they determine the number of variables we can measure and therefore, the degree of uncertainty in our measurements.

3. Can degrees of freedom change in a quantum system?

Yes, degrees of freedom can change in a quantum system. This can happen when energy is added or removed from the system, altering the number of ways the system can move or change.

4. How do degrees of freedom affect the energy levels of a quantum system?

The number of degrees of freedom in a system directly impacts its energy levels. In general, the more degrees of freedom a system has, the more energy levels it will have. This is because each degree of freedom can contribute to the total energy of the system.

5. What is the significance of degrees of freedom in quantum computing?

Degrees of freedom play a crucial role in quantum computing as they determine the number of qubits (quantum bits) that can be used to represent and process information. Increasing the number of degrees of freedom in a quantum system can lead to more powerful and complex quantum computers.

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