How old is the oldest particle or living cell in the human body?
Is any part of me left since I was born, or am I a completely different person?
How long would it take for a majority of the cells in me now to die off and be replaced by new cells?
When most of my body cells are replaced by new cells, can I blame my prior bad acts on my former self? 
I think a majority of your brain cells will have been with you your entire life. Few portions of the brain are known to replace existing cells
I have heard that fat cells are there for the long haul. Once they are made, they are not consumed entirely. Lipids will be emptied, but the cell will remain there, deflated.
I have even heard from a handful of people that the calcium in your bones is not replaced, but I think that is BS, because your bones are totally replaced every 10 years, I would imagine the calcium is as well. You can definately increase the density of your bones through physical means (playing tennis, raquetball, squash, or doing weightlifting, etc), and even older people can increase bone density, so I call BS on this last one.
Some of the protons (hydrogen ions) inside you are 13.7 billion years old.
Depends how you define “a” particle. Though the nucleon-number operator may not have changed eigenvalues in that long, how can you say that “this” proton is “the same” proton as it was yesterday?
Further, how can you say it’s a “proton”, since the same particle you point to must be seen as a neutron by someone looking with a different orientation in isospin space?
Fortunately, I didn’t pick “a” proton. AFAIK, there’s no way to reliably label the things, so you can’t tell how long a particular proton has existed. However, most of the protons around now formed shortly after the big bang, or we wouldn’t be talking about them having a half life of 10[sup]35[/sup]+ years.
Interesting replies. Anyone wanna take a crack at the question about how long most of my current body cells will remain in my body?
I find it fascinating in a way that in 10 years (?) the stuff that currently constitutes “me” will be gone and completely replaced by new stuff that constitutes “me” yet, I am still “me” despite being replaced by all new material.
I guess it’s like buying a car and slowly, over the years, replacing each part as is wears out. Eventually, no part will be original, yet it’s still considered the same car. On the other hand, if you bought a car, took it completely apart, dumped the pieces in the trash, bought all new parts, and assembled them, nobody would say the new car is the same car as the old car.
IIRC, if you’re female, you were born with all the egg-cells you’ll ever have.
Maybe.
That’s long been the accepted fact, but last year a study found that adult female mice could grow new eggs. It was pretty mindblowing, and shows us just how arrogant we are in our assumptions. Here’s an article about it.
But that’s not what’s known. To say a proton has such a half-life is a statement about the probability of the proton number operator changing eigenvalues. That is, a proton could vanish while another is created somewhere else. The total number stays the same, other than possibly one decaying every so often.
I understand that tooth enamel is not regenerated like most everything else: chipped teeth don’t grow back. So I would expect that those mineral atoms and molecules are largely there for the long run.
I believe the rest of the body exchanges pretty much all its atoms over the course of a few months (though, looking at it statistically, a portion of those atoms will remain indefinitely, rather like some atoms in a radioactive istope never decaying despite a half-life of years) - this figure of a decade seems to be pretty arbitrary in that respect.
As for the philosophical implications, it is indeed the arrangement of particles rather than the particular identity thereof which determines the macroscopic properties of anything from a dust storm to a computer. If we are a biological computer, then our memories are stored in cells which themselves regenerate. I suppose that literally speaking, I die every night and someone else wakes up with the same memories, rather like a computer compying its memory to a new drive every night before shutting down.
Because it has a positive electric charge. (Although I’m not sure if you’re just making a joke here.)
All experimental proton lifetime measurements come from watching lots of protons for lots of time and looking for any sensible final states. To the extent that the OP is happy assuming the standard model with a healthy choice of GUT extensions, what’s so bad about invoking proton lifetime?
No, I’m not. The proton and neutron (actually, the up and down quarks) are – according to the standard model – identical under an isospin rotation. What one person sees as a proton another may see as a neutron, just as surely as what one person sees as a circle another may see as an ellipse by turning his viewpoint.
Again, what has been looked for is a specific decay, not protons jumping from place to place but leaving the overall “number of them” (eigenvalue of the number operator, really) constant.
If we interacted with protons solely through isospin-invariant forces, then protons and neutrons would be indistringuishable. But our most familiar interaction with them – electromagnetic – is not isospin invariant. We can very easily distinguish protons and neutrons! (Even if we chose to ignore our electroweak abilities, the strong force is still only approximately isospin invariant.)
Are you just saying that no one can say that protons aren’t leaving one’s body via some teleportation-esque mechanism? While I agree that no experiment has looked for that, the OP is surely being less philosophical (that is, no “What can we know?” implied) and is presumably content with assuming standard-model-based mechanisms for proton transport.
No, I’m asking that how can we say that a particular proton is the same as it was even a minute ago rather than having been swapped with another one.
Okay, I admit, it’s all rhetorical. The answer is that we can’t. “Particles” simply don’t have individuated identities, and the standard model is essentially field-theoretic to begin with. What we nowadays casually refer to as a “particle” is a “basic excitation” of a quantum field, usually one with compact support. There’s really only one field, though, and you can’t meaningfully ask questions which presuppose individual identities to what are essentially bookkeeping tools to describe that one field.
In short, the question “how old is this proton?” is meaningless since there’s no “this proton”.