# Nature of the quantum degeneracy pressure

It is commonly known that it is the electron degeneracy pressure that prevents the collapse of a white dwarf into a neutron star, and it is not the electromagnetic force. However, it is also widely accepted that all the forces in the universe are just manifestations of the four elementary forces in the universe. So can the degeneracy pressure, caused due to Pauli's Exclusion principle, be shown to be due to one of these forces? I have read some other questions in SE How does the Pauli exclusion principle create a force in degenerate matter?, Degeneracy Pressure, What is it?, and they do not answer my question fully. In one of the links, the degeneracy pressure is described as an 'entropic force', but as far as I understand, all entropic forces can be explained by the underlying fundamental forces. If the degeneracy pressure is something that is fundamentally different from the other forces, why isn't it classified as a separate force?

## 1 Answer

The degeneracy pressure is indeed due to one of the four fundamental forces, but it takes a bit of though to see why.

If you put fermions into a box then their energy levels are quantised into the usual energy levels for a particle in a 3D box. So the first two fermions go into the ground state, then next two into a higher energy state and so on. Adding more and more fermions requires more and more energy giving us some total energy for all the fermions. It's the variation of this total energy of all the fermions with the volume of the box that determines the pressure. Note that this argument has assumed nothing about the particles other than that they are fermions, so no charge is assumed and therefore none of the four fundamental forces have been invoked.

Whch is all very well, but we then have to ask what is confining the fermions to the box? That is where the fundamental force comes in, because it's the gravitational force that is confining the fermions. Without gravity our gas of fermions would obviously just expand indefinitely to zero density and pressure. So gravity is the force ultimately responsible for the electron degeneracy pressure.

A degeneracy force is always due to some form of confinement. For example in ordinary matter the exchange force arises because electrons are confined into atoms due to the electrostatic forces.

• Very interesting! I m slightly confused: the gravity would still be there even without the box. Isn't the quantisation of energy levels alone behind the confinement here? – user929304 Nov 11 '15 at 16:29
• Interesting explanation. I'm still a little uncomfortable with it, however. Saying "So gravity is the force ultimately responsible for the electron degeneracy pressure" seems strange contrasting with, "degeneracy pressure is what counteracts gravity in a white dwarf" (e.g.). With a normal star (e.g.), one would say that electromagnetic repulsions are what resist gravity, but one probably wouldn't say that 'the electromagnetic force is caused by the confinement of gravity' (which is none-the-less true, in the same way as your explanation). Is there a better way to think about it? – DilithiumMatrix Nov 11 '15 at 16:39
• @user929304: energy levels are only quantised in a confined system. Free particles do not have discrete energy levels. The gravity confines the fermions, i.e. creates a potential well in which they sit, and as a result the fermions have discrete energy levels and therefore a degeneracy pressure. – John Rennie Nov 11 '15 at 16:47
• @DilithiumMatrix: the collisions have nothing to do with electromagnetism, they just generate a force from the change in momentum. – John Rennie Nov 11 '15 at 17:54
• Raise the temp enough and they'll be strong force scatterings. That doesn't mean we'd say the pressure is due to the strong nuclear force. – John Rennie Nov 11 '15 at 17:58