...to stay within the bounds of realism...
We've had a bit of frustration over the increasing number of users who visit Stack Exchange asking us to explain their fantastic idea with scientific accuracy. Unfortunately, there's a practical truth to your question that's also true in kind for 99.99% of all other such questions.
Telekinesis doesn't exist.
Without understanding the basic physics behind telekinesis, it's impossible to explain in any way how science could be used to create a technology to create it synthetically.
What are your options?
Arthur C. Clarke once suggested that any sufficiently advanced technology would be indistinguishable from magic.
This is your path to success. We're going to help you create some technobabble (which absolutely, positively won't comply with the science-based tag for the purpose of your question) that helps create enough suspension of disbelief that your readers will enjoy your story without being distracted by your explanation.
1. Methods of locomotion
- Gravity (works on any substance)
- Magnetism (only works on ferrous metals)
- Superconducting levitation (doesn't meet science-based, but could meet science-fiction)
- Quantum Entanglement (barely meets science-fiction because we're ignoring what it really is)
- "The Zoltan Effect" (brilliantly meets science-fiction because the effect isn't implausible, even though the physics behind the effect remain undefined)
It's worth underscoring "The Zoltan Effect." Author Frank Herbert used this literary device to rationalize his "shields" in the novel Dune. The shields worked due to "The Holtzman Effect." While he dropped a few tidbits about the Effect (and it's creator, Holtzman... great backstory there), he never fully explained it. He didn't have to. The Holtzman Effect wasn't in any way a critical aspect of the story. Spending time explaining it would distract from the story itself and simply give avid readers an opportunity to poke holes in his fictional "theory." Food for thought, there.
2. Methods of application
Once you choose the "force" or method of locomotion, we next need to decide how to apply that force. Since we're dealing with an artificial or synthetic solution, this means the "energy" isn't coming from the human mind, or even the human. It's coming from a machine. One that's hopefully small enough (ah... Clarkean Magic) to carry in one's pocket or on a wristband.
- Focused generator (e.g., optics like a laser or magnetic focusing, a "beam" of something is emitted from the generator)
- Displaced projector (combined effects resolve in a spot some distance from the machine, such as can be created with projected sound waves that combine at a distance to either cancel one another out or enhance and multiply their effects)
- Slave emitter (a machine not on the body, but located e.g. on the top of a building that then causes the effect).
3. Finally, the control mechanism
- Implanted cybernetics (electronics in your head that interpret mental commands)
- Headgear (a hat of some sort that contains the sensors to detect brain activity)
- Remote detection (a "beam" or similar sensing solution that is emitted from another location that detects brain activity).
A fun aspect of the control mechanism is that it's very likely to be an extension of pre-existing medical technology. Whatever "scanning" technology is used to identify problems with the body is adapted to "scan" the mind for control commands.
Please be aware for future questions that asking us to scientifically explain a completely fantastic idea isn't what we do. Anybody who could legitimately do that wouldn't post an answer here — they'd be running to the patent office with a multi-billion dollar solution. What we do here is help you rationalize your fantasy. Thus, in the future, please help us out by explaining with specific detail the effect you're trying to achieve including its limits, restrictions, and conditions, then ask us to what level of detail (not including "Real Life") you need the idea expressed to. We're happy to help! But it really helps us if you don't come with the expectation that everything can be explained through science. :-) Honestly, humanity is only just scratching the surface of what the universe has to tell us. Cheers!