Put concisely, No !
But, I feel it would be better for me to explain why in terms that non physicists can understand and if I do a sufficiently good job, you will conclude to yourself that the transporter, as presented in Star Trek, contradicts very well explored and proven laws pf physics.
First off, where did the idea of the transporter come from ?
Production budget costs of producing the TV series! Yup, they wanted to land the ships, but the miniatures and special effects would have cost more than their budget, so they needed to come up with a cheat. The
Wikipedia page on the Star Trek Transported explains:
Ok, no science or technical extrapolation there, they just needed a quick and dirty way to get characters between ships and planets.
Since I explained earlier that the procession of scientific advancement is a process of refinement, not replacement, just how good are the current theories ?
Well the pinnacle right now is
Standard Model of Particle Physics and it is sufficiently refined that it explains everything that you can experience in your life. It's failings include the origin of Mass and Inertia, but these have been hypothesized and experiments are underway to prove or disprove these hypotheses, Gravity, Dark Energy, and Dark Matter.
The Dark of Dark Energy and Dark Matter does not imply anything physics breaking, as some recent scifi's have alluded to, but instead to specific properties. In the case of Dark Matter, it means that what ever that Matter is, it does NOT interact with photons.
Want an example of Dark Matter that is fairly well explored ?
Neutrinos Yup, these guys are so dark that the only way to detect a few of them is to construct huge detectors under mountains and wait for the few interactions that do occur. They come from events that take place in the nucleus of atoms, the Sun, and every other Star, emits a constant shower that bathes the Earth and every atomic reactor glows in Neutrino emissions, a radiation that can not be shielded and that you can not hide from.

Yes

your body has atomic Neutrino radiation passing through it right now as you read this and there is nothing you can do to hide.

But, since Neutrinos don't interact with photons, light, and hardly interact with anything else, this is harmless and fine.
The Cosmology Dark Matter is the result of measuring how fast galaxies rotate. The speed is too fast so there has to be something there providing enough gravity to hold everything together. What that particle is is the present mystery and theorists have determined that it would have to be a lot heavier than Neutrinos.
The Cosmology Dark Energy is also a mystery, hence the use of the word Dark, but this came from the realization, from viewing ever more distant galaxies, that the Universe is getting bigger and something is making this expansion accelerate.
So, we know the Standard Model is sufficiently accurate to handle everything apart from gravity and a few cosmological scale phenomena. Lets get back to the Star Trek Transporter before we spin off on another tangent.
There is phenomena and quantum physics properties that appear paradoxical to our perception of the world. One being the
Heisenberg uncertainty principle which states that you can't know with certainty a particle's position and velocity. In later Star Trek series, the show's authors concocted an add on to the transporter to provide a sense of scientific accuracy, the
Heisenberg compensator to cope with this ambiguity, but it still falls short of the laws a physics.
Parallel with the Heisenberg uncertainty principle is the
Wave–particle duality which is commonly described as "all matter exhibits both wave and particle properties", but this just makes things confusing. A more accurate description is that all particles move as waves, but when observed they exhibit properties of being particles. The wave itself can never be observed and only hinted at when the path that the particle propagated though requires wave properties.
I'm not into sports but it will make a good metaphor - base ball. We have a bunch of people out on the field, the pitcher throws the ball and the instant that it leaves his hand, it vanishes. When held, this quantum base ball was a ball, but now that it's been thrown, it is a wave. All of the players hold out their mitts, including the pitcher, and after a period of time the ball is caught in someone's mitt. The wave has spread across the field. but due to probability found the most likely mitt and materialized back into a complete quantum base ball.
Here are examples of Gaussian probability curves of the quantum base ball:
A property to note that that a Gaussian curve will never reach 0 on the Y axis, it will get very close but it will never get there. That means even the rare case of the pitcher's own mitt could catch the very ball that was thrown.
Lets look at these curves. The Blue curve has a high peak and falls off quickly. This means that you have a far better idea where the quantum base ball is than you would with the Red or Yellow curves. All three peak at the same location on the X axis, which says the player whose mitt as at that location would have the highest probably of catching the quantum base ball, but since the probability does not reach 1.0, there is always a chance that it could end up else where.
Lets mix in a little more Wave–particle duality. The faster the pitcher throws the quantum base ball, the more spread out becomes the probability curve. That means the Blue curve is a slow ball, Red is faster, and Yellow should knock a player down.
If a player at location 0 on the X axis puts out their mitt and wow, there appears the catched quantum base ball, How fast was it going ? Well, the wave collapsed into the quantum base ball and all phenomena associated with the balls wave function and velocity are gone. When you know where the quantum base ball is, you know nothing of it's velocity, the Heisenberg uncertainty principle.
Now, lets get the velocity of the quantum base ball. To do this, you need to measure the probability function through out the whole playing field. If you measure at regular increments of distance, you'll find that you can never get "all" the possibilities to add up to 1.0, so you need to put in more and more sensors. Ok, lets say you were able to get the probability of the whole wave function to 0.999999. Well, now you have a really good idea of the velocity of the quantum base ball, but where is it ? When you know the velocity, energy, of the quantum base ball, you then have no idea where it's located, again, the Heisenberg uncertainty principle.
See, this is fun and easy.
So you can never know everything about every particle. Add to that the requirement that each proton and neutron requires 3 quarks, and for each proton you'll need an electron. To construct a person, you'll be needing a lot of particles.
Lets complicate the transported further. Supposedly, a person is taken apart and converted to energy, sent to the destination, and put back together. Well, what the hell is energy ? In quantum physics energy is combined mass and velocity. You can extract the velocity component by drawing off this component of the energy, but you can't do the same with mass.

Besides, as the velocity energy gets removed, the temperature of what remains falls. Even the energy release of atomic and Hydrogen bombs is feeble compared to the mass involved, yet the source of the explosion energy is actually the from reorganizing atomic nuclei to lower energy configurations.
Now it gets worse. In the Universe as it is now, quarks can't exist on their own and there is no evidence that a quark can be broken down further. So, if you could rip apart a Proton into it's constituent quarks, you'll have to add energy, lots of it - ( something that would make CERN's large hadron collider look small ) which changes the state of the matter. Then once you have isolated a quark, it could be freed to drift into the Universe according it it's wave function or kept as a particle, but then there is no way to put it where it needs to go, let lone reconstruct every particle of every atom that constructs a person.
Ok, your Dr McCoy worries about having his atoms scattered across the Universe. Lets turn him into energy by focusing a blast of energy of sufficient intensity to melt protons and neutrons into a quark-glueon plasma, *** BANG*** all the while noting the initial state of every particle and hope that you can get enough information to put things back. Then extract the resulting gain in liberated energy ***BOOM*** and what is left plummets to a few micro-kelvins in temperature.
I suspect that this would have the appearance comparable to a planet busting atomic explosion.
At the destination, you'd need to somehow deliver the raw material, electrons, quarks, glueons, and control the cloud as you condense it into something looking close to the Dr McCoy that you started with. Remember, that some information had to be lost when you took your measurements, so hopefully it wouldn't matter if there is a missing proton here or there.
Oh, and don't forget that what you do get must not only comprise the right assemblages of organic molecules, they have to be in a biologically viable state after the extreme energy exposures of ripping him apart and then putting a new incarnation together, It's Alive ?
