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I'm having some problems understanding what are the possible definitons of mass and how they are related to each other.

In Classical Mechanics, we can distinguish between inertial and gravitational mass (although they are supposed to be the same $m_i=m_g$): $$F=m_ia$$

$$F=G\frac{Mm_g}{r^2}$$

On the other hand, in Special Relativity, mass is: $$p^\mu p_\mu=m^2$$

But in Quantum Mechanics, I think none of these definitions should be valid (Newton's second law doesn't hold, gravity can't be described in QM and QM isn't relativistic). So, what mass should we use in Schrödinger's equation, when we state that: $T=\frac{p^2}{2m}$?

In QFT, the correct mass is $p^\mu p_\mu=m^2$, isn't it?

Finally, are there more possible definitions of mass, besides the description in String Theory (maybe in General Relativity, I've read that there are problems defining energy in a localised point, that we can only evaluate the total energy)? Are some definitons more fundamental than the rest?

EDIT: What about this definition I've found in QM $m=\hbar (\frac{d^2E}{dk^2})-1$?

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Related: physics.stackexchange.com/q/8610/2451 and links therein. –  Qmechanic May 18 '13 at 16:43

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The definition $m^2=p^\mu p_\mu$ is generally valid.

By the correspondence principle, the definition of mass in relativity has to reduce to the Newtonian one in the nonrelativistic limit, and the definition of mass in quantum mechanics has to reduce to the Newtonian one in the classical limit.

But in Quantum Mechanics, I think none of these definitions should be valid (Newton's second law doesn't hold, gravity can't be described in QM and QM isn't relativistic).

Newton's second law holds in the classical limit of quantum mechanics, and that's enough to define the mass of any given object. For example, in the Stewart-Tolman effect, we're dealing with a large number of electrons, and large particle numbers are one way in which the classical limit can be obtained from quantum mechanics. Once the mass of an electron has been established by such a classical technique, it's established in quantum mechanics.

It's not really true that gravity can't be described in quantum mechanics. Quantum mechanics does fine with gravitational fields, just not with spacetime curvature. (By the equivalence principle, you can have a gravitational field in flat spacetime.) For example, people have done neutron interferometry in a gravitational field (Colella 1975) and gotten exactly the results you'd expect from freshman quantum mechanics, without having to do any quantum gravity. This means that even in a quantum mechanical context, we can verify that inertial and gravitational mass are equivalent.

maybe in General Relativity, I've read that there are problems defining energy in a localised point, that we can only evaluate the total energy

Actually even the total energy can be impossible to define in GR. But that doesn't stop you from, e.g., defining the mass of an electron or a galaxy. There are conserved, scalar measures of mass that can be defined in any asymptotically flat spacetime.

The real issue in relativity, even in SR, is that mass isn't additive. Also, in GR, it's not mass that's the source of gravitational fields, it's the stress-energy tensor.

Colella, Overhauser, and Werner, Phys. Rev. Lett. 34 (1975) 1472

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