# What is a String Mass-Shell? and what does it do?

## Main Question or Discussion Point

I am researching superstring theory and have recently seen a equation called a "heteroic Superstring mass shell Condition". Can anyone tell me what it is and what it is used for. Thank you,
-Joeltopian

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I am researching superstring theory and have recently seen a equation called a "heteroic Superstring mass shell Condition". Can anyone tell me what it is and what it is used for. Thank you,
-Joeltopian
I don't have a clue! But since no one else has answered your question, I'll toss out a tidbit, in case it is actually the same thing.

In "normal" Quantum Field Theory, when people speak of a particle as being on its mass shell, what they mean is that its mass is equal to its physical mass, i.e. what you measure when it is observed. Particles can become "virtual" and acquire a different mass than what they are measured to have - for example, a photon can have a non-zero mass - when that's what's needed in order to conserve energy-momentum. The particle must then interact again to produce non-virtual particles, which are on their correct mass shells and can be observed.

The reason it's call a "mass shell" is that in relativity, the magnitude of the the four-momentum is just the square of the particle's mass. To say that a vector (in ordinary space, anyway) has a fixed magnitude is the same as saying that its tip can be drawn somewhere on the surface of a sphere of radius equal to that magnitude.

So, if a particle has a non-physical mass value for a short while, that's like saying that the tip of its four-momentum vector is no longer on the "sphere" whose radius is defined by the particle's physical mass - hence, it is "off shell".

Does any of this apply to strings? Beats me.

so basicly its the way to mesure a particles mass in quantum theory?

It's easiest to explain this step by step.

In special relativity (no quantum mechanics yet), the "mass shell" or "mass hyperboloid" is the set of all energy-momentum combinations for a particle which give the correct rest mass for that particle, if you transform to the frame of reference in which the particle is at rest. If a particle has rest mass M, and a particular state of motion (energy-momentum combination) was equivalent to having some other rest mass M', then you would know that this isn't a physically possible state of motion for that type of particle.

A "mass-shell condition" is an equation which tells you whether a particular energy-momentum combination is on the mass shell or not, and thus whether it's physically possible.

In quantum mechanics, particle states are represented by wavefunctions, which encode energy and momentum of the particle in their frequency and wavelength. So the mass-shell condition is now expressed in terms of the properties of the wavefunction.

Also in quantum mechanics, one of the ways that you obtain the probabilities of events is by summing over all sorts of possible histories (with particles going this way and that, transforming, emitting and absorbing, etc), with each history making a contribution to the final probability. Mathematically, these "sums over histories" are integrals. Integrals are often easier to solve with a change of variables, or by changing the range of a variable that is summed over. You might be trying to integrate a function over the real numbers from -infinity to +infinity, but it will be easier to integrate it over all the complex numbers, and you have some theorem which tells you that the answer is the same - that sort of thing often happens.

When you are doing the quantum sum over histories, and you change the integral to an equivalent one that is easier to solve, that can mean that the particles appearing in the middle of the history take unphysical values for their properties. That's the result of doing the integral over a broader range of values. This can mean that the "virtual particles" "go off shell". But the particles that are observed, at the beginning and at the end, have to be "on shell", because that part of the calculation directly corresponds to reality.

(There are also ways to do the sum over histories where you stay physical and "on shell" from beginning to end.)

Finally, when we get to a quantum string theory like the heterotic superstring... the energy of the string is in its vibrations. And since it is a quantum string, again we have a wavefunction, so it's a little complicated. But again, you have mass-shell conditions, which tell you what sort of string wavefunctions correspond to physically meaningful quantum string states.

So to sum up, a mass-shell condition is a technical statement about what sort of quantum wavefunctions correspond to genuine physical states; and for the purposes of calculation, you will sometimes use wavefunctions that temporarily go off-shell, because that makes it possible to calculate the integrals, but probably it's just a mathematical technique and not a physical reality.

(I say "probably", because there is a lot of disagreement about the physical reality behind quantum mechanics, and it's logically possible that there is an "ontological interpretation" of quantum mechanics in which going off-shell actually happens.)

Mitchell Porter said:
In special relativity (no quantum mechanics yet), the "mass shell" or "mass hyperboloid" is the set of all energy-momentum combinations for a particle which give the correct rest mass for that particle, if you transform to the frame of reference in which the particle is at rest. If a particle has rest mass M, and a particular state of motion (energy-momentum combination) was equivalent to having some other rest mass M', then you would know that this isn't a physically possible state of motion for that type of particle.

In "special relativity (no quantum mechanics yet)" there is only one "correct rest mass ", namely the M of a particle with no observable energy or momentum associated with its absent motion, in the "frame of reference in which the particle is at rest". In this context imagining another M' in that frame seems to me to be nonsensical. The terms "mass shell" or "mass hyperboloid" are here inappropriately borrowed from quantum mechanics where there are still mysteries, like the Uncertainty Principle that allows virtual unobservable particles to be imagined.

But only when such imagining makes possible quantitative and predictive descriptions. Not when clarifying a puzzling concept.

Paulibus, the mass hyperboloid is not a concept specific to quantum mechanics. It is the set of all energy-momentum four-vectors which are relativistically equivalent to a particular four-vector (rest mass, 0, 0, 0), where the last three zeroes indicate zero momentum, i.e. the particle is at rest.

See the curving lines in this diagram (the boundaries between green and white areas). Each of those boundaries is a "mass shell" or "mass hyperbola", for a particular rest mass. That means that a point lying on the hyperbola is a possible energy-momentum vector for a particle of the corresponding rest mass. If a particle has such an energy-momentum vector in one frame, there will be another reference frame (related to the first by a relativistic coordinate transformation) in which its energy-momentum vector is of the form (rest mass, 0, 0, 0), the vector at the lowest point on the hyperbola. Conversely, if a particle is known to have a particular rest mass M, and if a given energy-momentum vector can be transformed to (M',0,0,0), M' not equal to M, then that energy-momentum vector is not a possible real-world state for that type of particle, because it's on the wrong mass hyperbola. That's what I was saying.

In general being 'on-shell' in position space means that the equations of motion are satisfied. For a plane wave solution to a free-relativistic particle (to the KG or dirac equations) the Fourier transform to momentum space implies p^2=m^2 where p is the four-momenta this is what is meant by being on the mass shell for a particle. Not sure how these generalises to string theory.

Thanks, Mitchell Porter. That's much clearer. Allow me one last quibble: replacing your "are relativistically equivalent to a particular four-vector (rest mass, 0, 0, 0)" with "have the same magnitude as the four-vector (rest mass, 0, 0, 0)" may be simpler.