A question about renormalization

Typically in quantum field theory the values of quantities are adjusted to compensate for infinities. Would it be possible instead to alter the definition of zero to compensate? Could zero be defined as a constant below which existence is undefined, based on the Planck Constant? Could that definition help define infinity and 1/0?

So if we alter the definition of zero, does that mean that I currently don’t have zero eggs in my fridge, but {infinitesimally small number} eggs?

Maybe you should explain what you mean. In physics, no infinite quantities are rigorously considered, only well-defined finite quantities. The short answer (or sketch of an answer) is, classically (as classic as quantum field theory, anyway), you pick an “ultraviolet cutoff” parameter (a positive real number, not infinity) and define cutoffs of your perturbative QFT, which let you introduce effective S-matrices and “counterterms” allowing you to find a UV regularization and take the limit as the cutoff goes to infinity.

So the procedure involves this cutoff, but at no point is any number equal to infinity.

Well there is a very small chance that particles have spontaneously arranged themselves in the shape on an egg, and those waveforms are just overlapping into a percentage until you open your fridge :cool:

The first step to answering your question is to learn calculus. Limits allow us to do things that look like dealing with infinities, while only ever actually dealing with finite numbers. In the process, we render it completely unnecessary to ever actually divide by zero.

You asked exactly the same question in this thread a whole three days ago, in which you were told to do some reading about infinities. Did you?

Moderator Note

Please do not junior mod (and especially, don’t be snarky about it). The OP sent me a PM requesting further discussion of the question and I gave my permission.

General Questions Moderator

I understand limits. But does mathematics (theoretical) accurately describe physics? Could 1-0.999… be defined as a constant (the lower limit to measurement below which existence is undefined). Then 0 would be defined (wrt physics not theoretical mathematics) and infinity =1/0.

What do you propose should be in fact infinite? We can talk about the gravitational potential due to a point mass, or about point particles in quantum mechanics (e.g., electrons), or in mathematics about the delta “function”, but what problem do these things present?

Where you do have to do a lot of mathematical work in physics is (for instance) to deal with things like unbounded operators, and books have been written about that subject and about other mathematical underpinnings for quantum theory, but how is exchanging the real numbers for something else going to simplify anything?

Any given experiment has some lowest value it can measure. But it’s different for every experiment. No purpose would be served by attaching some fundamental label to whatever happens to be the best experimental bound we’ve so far achieved.

I read up on Planck units. Even if you defined 1m - 0.999…m as 1.616255(18)×10−35 m, and the maximum diameter of a circle as the inverse, physics already uses Planck units and this wouldn’t help with renormalization. I apologize for not further investigating natural units before I asked my question.

And there’s no particular reason to assume that the Planck length is the smallest length possible, anyway. Or even to assume that there is any smallest possible length. Maybe length is quantized, and if it is, maybe it’s quantized in such a way that there’s a smallest length, and if so, maybe that smallest length is somewhere in the vicinity of the Planck length… but those are all maybes.

The limit as n approaches infinity of (1+1/n)^n = e seems to suggest there is a lower and upper limit to the ability of something to exist, also implying a lower and upper limit to the universe. If a lower theoretical limit to existence below which existence is impossible was discovered, and the inverse is the upper limit beyond which existence is impossible, singularities would not exist and there would be an upper limit to the diameter of a circle and the maximum extent of gravitational influence. Could this help reconcile GR and QM?

Physicists can use math to understand and calculate properties of things, even basic things (matter, energy, time, information, etc.) But math doesn’t define physics or set any boundaries per se any more than English defines biology. You cannot change a human by redefining the word “body.” Mathematical limits do not imply anything about the existence or non-existence of physical objects. That’s a non-starter.

Energy scales (or length scales) are important in physics. If you had to worry about quantum fields or grand unified theories or strings when modeling, say, continuum mechanics, you would have a problem.

No, you are misusing “limit”, which is a short-hand term for “point of convergence”. This is why my HS calculus class began with a bunch of kind of boring rote involving series.

The most accessible example of convergence might be the manual calculation of a square root: you start with b=2 and do a = (b + x / b)/2 , check to see if |b - a| is still greater than your accuracy requirements and, if it is, set b = a and do it again until you get close enough (or reach 0, if the square root turns out to be an integer). It might be expressed using lim( |a - b| -> 0 ).

So, renormalization is simply encountering an “infinity” (short-hand for “non-useful value”) in your math and reframing the expression in terms of a convergence so that you can extract something useful.

Posts like this are why I say you need to learn calculus. If you understood limits, then you would understand that they imply nearly the exact opposite of what you say they do.

I remember endless exercises in Calculus 101 like lim x => 1 of (x^2 - 1)/(x-1). Factor and x=2. So now you can understand the area of an infinite number of shapes and the rise/run of a point on a curve. But with respect to e, what is lim X => infinitely of 1/x?

But do you remember what a limit means? The whole point of a limit is that no matter how small a number you name, I can always name one smaller.

Are you saying that the lim x => 1 of (x^2 - 1) / (x - 1) = 1.999… ?