In my idea for a story, my main character has Telekinesis. My problem is, this is a definite hard science fiction world, and he is abnormal in the fact that he is the only known one who can perform Telekinesis. How would I scientifically explain his abilities? I am fine to bend or even break currently accepted laws, but I do not want to introduce magic into the equation.
I'm no physicist nor am I an expert in quantum mechanics, but I think I see a way to combine some comments and @anon 's answer to give it a fig leaf of plausibility.
The major problem is Conservation of Momentum. If you want to move something you have to apply enough force to overcome inertia, friction, and so on. That force has to come from somewhere.
Here is where your "telekinetic" gets the necessary force. He can sense and link quantum entanglements with his mind and can link objects together. For example he senses the rock in his hand and the pencil on the table. He links them, and when he drops the rock, the pencil goes flying. Or he can get creative and use the falling rocks from a landslide and "borrow" some of that momentum and redirect it into the person who fell off the cliff with the rocks and save them.
This is absolutely not hard science, but I hope that it gives a quasi-rational explanation for the telekinetic ability aside from "Just because", or "She's a mutant and one of the X-Men"
Telekinesis is currently physically impossible.
As part of this mythology you need to exert a force remotely, as in no direct path from emitter to receiver.
There in lies the problem, no force in physics can pass through an object and then suddenly change direction and take action. Its always either a push or pull.
You could take advantage of the ambiguity of quantum physics. Have your user focus on a spot in space and concentrate quantum particles on that spot to create and destroy exotic matter that results in the effect of telekinesis.
Though that is a hell of a stretch and sure to put some people in an uproar.
Only move what you can actually move.
You could work around some of the lawbreaking by having telekinesis involving only forces the user can himself generate with his or her own body. I can lift a bowling ball using my muscles. I can lift a bowling ball remotely using those same muscles. The lifting of the ball creates an opposite reaction through my body and my feet push the ground harder. I burn calories of energy to do it. So too if I telekinetically lift the ball.
I cannot lift an elephant and so I cannot telekinetically move an elephant. If I try to push an elephant hard it will probably notice, but I am liable to push myself over backwards. So too if I try to telekinetically move something too heavy to move. If I try to telekinetically prevent a weight from landing, I take that force through my body. The weight would crush me down, even at a distance, until I chose to break the telekinetic connection. No Xwings coming up from the swamp with this brand telekinesis.
This would make the limits of what the telekinetic can do more clear. One could gauge how much effort a given telekinetic endeavor would expend. The new power, in addition to action at a distance, would be the ability to continually apply acceleration to a body - imagine pulling a boat on a rope. You can add energy to the boat incrementally according to your traction and strength. One can build up large kinetic forces with that sort of incremental addition. I can lift a bowling ball at a certain rate until my arm is fully extended and then I must stop. If not limited by my arm and I can continue to move the ball at that rate, it can get going very fast over a short period of time.
One could have the telekinesis be by virtue of perceiving a fourth dimension, so my hand even though at a distance in 3d is actually on the bowling ball because I can reach through the 4th dimension to adjacent space. This also means my hand be burned or otherwise hurt by the object I am moving at a distance. This is a method to have action at a distance without breaking any laws of physics - multidimensional space is not prohibited by physics.
I have thought much about this question as I created a RPG where I wanted psionics habilities to be done in a "realistic way" so they never get to a scale where they are so powerful that they actually become unbalanced with the rest, like high-level magic becomes in DnD compared to warrior abilities. Actually, my conclusions are that for telekinesis, it is quite difficult to work a balanced solution. In this process, I came up with question in power definition that might help you;
-Do you have to see it or do you just have to feel it thought the channel that allows you to manipulate the energy you use to perform your ability?
-Do you exert a push on a surface, so do you generate a force, or do you redirect forces or do you move the whole object?
-What is the extent of the push, or the tininess of the surface you can push?
-What is the weight you can lift, or the volume?
-What is the power, work or intensity of the forces you can affect?
-Do you generate one force at a time or can you generate more (it is important for movement, as it might be more complex/difficult to make something you lift rotate at the same time, or that's something you need to know if you want to tear something instead of pushing through it) ?
-Is the force you generate respecting the laws of Newton (when you stop using your ability, does the object conserve momentum, if you "push" the object by emitting a force through your hand, if the object is heavy, are you pushed back?). -If you generate the force, is it at any point of space and can you focus it just by wishing that this thing is moved, or do you have to focus accurately and thus this is made difficult by moving targets?
-has it a reach or is it unlimited? is it constant power on its reach or just decreasing like if you were emitting a wave that disperses/fades with distance?
-does you telekinesis allow you to use it as a grapple, lifting you by hanging something upper then tracting you, or are you able to fly or lift yourself with it? -Do you control it finely, exactly, do you have to focus? If you try to lift a glass, can you break it by applying power? can you miss and lift the liquid inside instead?
You have to question every aspects of power usage by wondering how it would work with various situations (as i did above) Answering these questions will help you defining requirements, or respect requirements you have in your anticipated storyline. It could also help you finding plot points and challenges for your character. Because I suppose you need some challenge, and a character that would have a telekinesis form that allow not seeing things to act upon these would have no challenge with doors, or locks, for example. Also, the idea of push or not is important, targeting too. Because if character happens to be fighting, if he just needs to thing to say, crack the skull open or push on the brain/internal organs of opponent without seeing these, or if he can push on the eyes (even if there are protection glasses in front of them)...
To answer more precisely, I could go for some kind of biomagnetism, or some implanted device (with an explanable or not mechanism, depending on its origin -governmental, extraterrestrial, but the black box attribute goes difficultly along with hard science), but that leave the question of energy and Newton's law, which might or not be convenient or challenging for your character, or for you to be done properly. Character may also generate a force that can distorse space without affecting the object in this space (much like a gravity wave), but allowing you to move it with the "tide". You also have the option for creating a new form of unknown forces with special set of requirements (much like magnetism only apply to metals), and properties and limits. Don't forget limits.
You can also go for the nanotech option, but that's the least interesting for me, because while seeminly hard science, actually that would be nanotechnobabble instead (just like kryptonite being a cause for any power besides superman's in smallville, nanotech is so much used these days it became a used rope trope). Using some for of connection between objects (like the quantum entanglement solution) allows to define easierly the limit of your power while some aspects need to be more precisely defined (do you need some vector or can you link your body and objects directly).
Also you have to define the energy needed, if character only needs to think, if it can be a reflex or if he can use it only by complex calculations or manipulation consciously, if it exhausts him, or if it is like running, you can do it as long as you want or close to it depending on the intensity, or do you have some "fluid" that you run out (temporarily or permanently).
To call out a solution I read, brain waves are not waves, except on ECG oscilloscope. They just show electrical signals evoked in your neurons...
Gravity is the force that can move objects. If there were some way to create space-time curvature, or if you want to expand on the Graviton theory, you could come up with a way that allows for something similar to telekinesis, and with its own restrictions and rules.
The main difference between most magic in literature and science is not so much the explaining part, but that most writers use magic as a cheating tool. Not many take the effort to actually create a consistent system of magic with rules, limits, requirements.
If if the scientific theory behind your telekinesis is total hogwash, the fact that there is a theory and that the abilities, strengths, weaknesses and limitations of telekinesis can be logically deducted from it already sets it apart from most magic descriptions and lifts it into a realm where suspension of disbelief can turn it into science "fact" (within the story) in the mind of the reader.
Instead of giving him telekinesis give him limited control of gravity. It wouldn't explain how he can do theses things. But it will explain why things can floor in mid air.
With all the other problems people have stated, let me add this one: energy.
Even if it was possible to apply force remotely, where does the energy come from?
Take a look at: How might telekinesis and pyrokinesis work if they were possible? here from a year ago.
And: This wiki
The one thing that I saw that I cannot find on line is that the heat generated while producing the power to lift a 1 gram object against gravity would fry someone's brain. I really wish I could find a source for that. Sigh.
So, unless you can figure out where the energy comes from, you might be out of luck.
It looks like you will have to hand wave something but, maybe, you can push the hand waving out far enough that most people won't see it.
You can't use psionics in hard sci-fi, if you do it ceases to be hard sci-fi.
Psionic abilities are just magic with a slightly different explanation...that is no more scientific in actuality to explanations of magic in fantasy as manipulation of magical energies.
If you use some sort of high technology to produce the effect (an implant with a graviton manipulator) then it is no longer telekinesis.
My recommendation is that you relax the requirement for hard sci-fi.
Hard science: No.
That said, the best theory I can give you is that humanity has always had the potential ability to manipulate reality—something intrinsic to our intelligence probably. To activate the ability requires the right mindset, and maybe the right genes. The right mood or mindset creates the right brain chemistry to allow the ability to turn on, but then it also takes the right effort or skill to actually apply the ability.
You can cite all the scriptural miracles as evidence. If your character isn't a prophet sort of person, then maybe his genes are a new mutation that allow for a wider range in brain chemistry to turn on the ability than the rest of humanity.
The apparently floating object is actually being lifted by an unseen swarm of flying, remotely controlled nanobots.
Your abnormal protagonist simply has early access to this cutting edge technology, which will eventually reach mass marked. He's possibly the inventor, or very close to the inventor. Married to the inventor maybe.
He can vocalize the resonant (inaudible) frequency of objects to levitate them and move them by independently varying the pitch of each vocal cord. (similar to this youtube demo)
Alpha Wave Tractor Beam
Similar to above, but using brain waves.
Genetic mutations caused him to be able to control very strong electric currents (like an electric eel) and turn them into magnetism using his ferromagnetic iron rich hemoglobin (also a mutation), thus creating a stronger form of biomagnetism or bioelectromagnetics
I think one possible way for someone to perform telekinesis (which isn't mentioned in these previous answers) is for them to be able to generate an extremely strong static electric field. People commonly can accumulate enough static for non-metal and non-magnetic objects to be attracted to them, and I think it's worth mentioning that the molecular machinery in the human body might be better suited for generating strong static fields, rather than magnetic fields. An individual human DNA strand has about a 10^-9 Coulomb charge, and there are trillions of cells in the human body, so the amount of charge in the human body from DNA is very large, it's just balanced out. But maybe a telekinetic character has a charge imbalance, which could result in a very large static field. This could have telekinetic effects, although this character would probably have hard to control telekinesis. Maybe the strength of their field could be determined by their diet?