Are matter waves transverse and can they be polarized? Are matter waves transverse and can they be polarized?
What I know:I'm aware of the de Broglie matter waves hypothesis and de Broglie wavelength relation(at a very basic level as part of high school curriculum). But, it is not mentioned anywhere about whether they are transverse or longitudinal. If they were transverse they could be polarized but are they?  
I could understand that matter waves are not real waves and therefore there is no case of longitudinal or transverse. I know that the square of amplitude of the wave at a position gives the probability of finding the particle at that position but, why is then frequency defined for matter waves(in other words what is its use?)?
 A: Different possible polarizations of a "matter particle wave" corresponds to the different possible degrees of freedom of the quantum field describing the "particle".
For a photon, we have 2 possible polarizations (for instance : vertical polarization, horizontal polarization). For a electron, we have also 2 possible polarizations (for instance : left handed, right handed). For the positron, we have also the same 2 possible polarizations , and the whole electron/positron quantum Dirac field describes 4 possible polarizations.
However, transversality has to do with a precise space-time condition, and this notion is only available for some Lorentz representations. A transverse relation will be written :  $\vec k.\vec \epsilon_\lambda (k) = 0$. However, it suppose that the Lorentz representation  of the field is a "vector", which is (roughly) true for the photon field, but false for the electron/positron Dirac field. In the latter case, the representation is a bi-spinor, so you cannot get a transversality relation directly between the momentum $\vec k$ and a bi-spinor like $u(\lambda, \vec k), v(\lambda, \vec k)$ (you will have to involve bilinear (quadratic) quantities based on bi-spinors to get "vectors").
In the same way, the notion of longitudinal wave $\vec k$ parrallel to $\vec \epsilon_\lambda (k)$, is a nonsense in the case of the Dirac field.
A: I will reply to this:

why is then frequency defined for matter waves(in other words what is its use?)?

Frequency is defined for electromagnetic waves. When the photon was discovered and the theory assigned to it an elementary particle identity, on par with electrons and protons (at the time ) it was found that the frequency of the electromagnetic wave which was composed by many particles coincided with the energy of the photon times a constant E=h*nu . In this case, there is no question why the same frequency is used.
When matter in the micro world  was postulated to act as a wave, and it displayed a frequency in the two slit experiment the analogy with the photons was glaring, presuming one was building up a model of nature in a unified manner, and thus De Broglie connected the energy/momentum of a matter particle to the observed wavelength.  It worked as a first level model.
When quantum mechanics was formalized with its postulates and its equations it became clear that the diffraction pattern from the (quantum-mechanical-dimensions) particles , photons, electrons,.. was a probability distribution that could be calculated and predicted by the boundary conditions on the solutions of the equations.The probability  dislays  interference patterns.
Thus the identification of matter with a frequency allowed to describe data, and predict future behaviors in a unified and concise manner.
Its "use" is similar to any factual number on particles at the quantum level: to describe existing data and predict future behaviors.
A: I think that matter wave could be both longitudinal as well as transverse!!!
Even if, by this time, any scientist anywhere proved that it is either longitudinal or transverse, may be corrected by somebody else in future with this change - it is both longitudinal as well as transverse.
As we know generally, the concept of matter wave is just the idea of probability of finding a matter at a particular position as the square of amplitude of the wave. Then, this probability is equally valid with a longitudinal or a transverse wave. 
Another fact we all know is that, thoughts come first and then it changes into actions and results. In the same way, we could say that in the universe, field and waves could be the first existing ones and then came the matter. A wave is a disturbance produced in a medium and hence, the field itself could be the first medium, with equal probability for a longitudinal and a transverse wave. Both may not be existing simultaneously. However, since the speed is too high, even if we look into the origin of the wave itself, we may not be able to distinguish correctly the time delay of production of these two different types and we may feel that both are simultaneously existing. Matter produced from longitudinal waves (or the probability of finding a matter in the longitudinal waves) and matter produced from a transverse waves (the probability of finding a matter in the transverse wave) could have different properties or characteristics. 
These topics need further research and I am sure that just the concepts of physics would not be enough to reach the result - In physics we know about the fields, forces and all. As an example, we know that the like fields from a charge or magnet will repel each other. We do not know two fields, interacting with each other, changes its nature (from attraction to repulsion or the reverse) due to the circumstances or surroundings. However, we all know that, our thoughts and attitudes, which are also nothing but fields, created by the complex brain neurons and fields, will change the nature. Same thing could be a matter of love or attraction for the same person at one time and may be a matter of hatred at another time. 
Instead of going ahead through a single branch of science, we should think of joint researches by experts from different fields.
