# Thread: infinity equals negative one half?

1. alright, my calculus teacher just told me that . This also has something to do with the Riemann Zeta Function(no clue), and I'd rather stimulate conversation here than search wikipedia. Can anyone please tell me why he chose to kill my brain?

2.

3. Also

Of course these are not genuine equalities. They are motivated by the Riemann zeta function:

If we analytically continue , it turns out, for example, that . Of course, near -1, the zeta function has a completely different definition than the series definition given above. However, if we formally let s=-1 in the above equation, we get the infinite sum 1+2+3+4+...

Associating infinite sums with values of the zeta function in this way might seem like a silly game, but there's actually something deep going on here, which I don't fully understand. String theorists routinely encounter diverging sums of the above types, and they get around these infinities by using this zeta function trick.

I know it's tempting to dismiss this as nonsense, but this regularization procedure ends up producing some very interesting and non-trivial mathematical objects. To be more precise, if one computes certain string partition functions, and uses the zeta function to get rid of infinite sums that crop up, the partition functions end up being modular forms.

4. that's what my teacher said, but never gave me a reason, and left me with the helpful advice, 'if you really want to know, go look it up!' and he said this because he didn't have enough time to explain it, as it would take a while. So, now, I'm genuinely curious and would LOVE to know why, without making my head explode, if possible.

5. Originally Posted by salsaonline
Also

Of course these are not genuine equalities. They are motivated by the Riemann zeta function:

If we analytically continue , it turns out, for example, that . Of course, near -1, the zeta function has a completely different definition than the series definition given above. However, if we formally let s=-1 in the above equation, we get the infinite sum 1+2+3+4+...

Associating infinite sums with values of the zeta function in this way might seem like a silly game, but there's actually something deep going on here, which I don't fully understand. String theorists routinely encounter diverging sums of the above types, and they get around these infinities by using this zeta function trick.

I know it's tempting to dismiss this as nonsense, but this regularization procedure ends up producing some very interesting and non-trivial mathematical objects. To be more precise, if one computes certain string partition functions, and uses the zeta function to get rid of infinite sums that crop up, the partition functions end up being modular forms.
Nobody fully understands it. The process of renormalization that physicists use in quantum field theories is a total mystery from a mathematical perspective. It results in astounding accurate predictions of experiment, but make no sense mathematically.

Neither does, analytically continuing a function, evaluating it at some point and equating it to a power series that does not contain that point in its radius of convergence.

There is, as you say, probably something quite deep going on, but if you can describe what it is you may be on your way to a Fields Medal or Abel Prize.

Anybody who presents this stuff in an elementary calculus class without further explanation ought to be horse-whipped.

6. I wouldn't go so far as to say that it makes "no sense" mathematically. The procedure itself is well-defined. Start with an operator A. Let be the eigenvalues of . Then, when confronted with the series

,

interpret this as the analytic continuation to of the zeta function associated to this operator, that is, of the function

Of course, if the analytic continuation has a pole, then you need to do more work, and I'm not sure what the physicists do in that case. Also, you need to make sure the given function is indeed meromorphic. But these technicalities aside, the procedure is well-defined mathematically.

A more interesting question is what is the physical meaning behind this kind of regularization. My sense is that in some instances at least, this is also understood. See , for example, A Zee's discussion of the Casimir effect in "Quantum Field Theory in a Nutshell".

7. Dr. Rocket,

It results in astounding accurate predictions of experiment, but make no sense mathematically.
I've always believed that there is a link to anything that can be described mathematically and our ability to reason, (a sort of "mental proof"), and that we can use the former to derive the latter. In general however, scientists tend to cheat this by conducting expirements and not having to settle for the more error-prone reasoning (although there is reasoning behind this concept as well). Do you believe this applies to this particular case (that is, Reimann's Zeta Function suggesting that ), and if so, all others?

8. Originally Posted by salsaonline
I wouldn't go so far as to say that it makes "no sense" mathematically. The procedure itself is well-defined. Start with an operator A. Let be the eigenvalues of . Then, when confronted with the series

,

interpret this as the analytic continuation to of the zeta function associated to this operator, that is, of the function

How does A come to have a guaranteed logarithm and why is the exponential that you suggest of trace class ?

9. okay, this all is very confusing to me, but I must ask, what in the name of all that is holy is ?

10. Originally Posted by DrRocket
Originally Posted by salsaonline
I wouldn't go so far as to say that it makes "no sense" mathematically. The procedure itself is well-defined. Start with an operator A. Let be the eigenvalues of . Then, when confronted with the series

,

interpret this as the analytic continuation to of the zeta function associated to this operator, that is, of the function

How does A come to have a guaranteed logarithm and why is the exponential that you suggest of trace class ?
Come on man, now you're just trying to bust balls for no reason. Obviously one has to assume that the operators given are sufficiently nice so that the above formula makes sense. Which presumably is the case for interesting examples that arise in physics, in which case, if memory serves me, A is some nice second order differential operator on a compact manifold (like on a circle).

If you want to play the nit-pick game, you might as well object every time someone writes an equation like:

After all, what if x is zero, and which branch of log did you pick? Unless you want people to spend entire lifetimes composing on this forum, you're going to have to get used to a slightly looser style of exposition.

11. Originally Posted by Arcane_Mathamatition
okay, this all is very confusing to me, but I must ask, what in the name of all that is holy is ?
I assume you are writing for sqrt(-1). Anyway,

pretty much by definition.

12. Originally Posted by salsaonline
Originally Posted by DrRocket
Originally Posted by salsaonline
I wouldn't go so far as to say that it makes "no sense" mathematically. The procedure itself is well-defined. Start with an operator A. Let be the eigenvalues of . Then, when confronted with the series

,

interpret this as the analytic continuation to of the zeta function associated to this operator, that is, of the function

How does A come to have a guaranteed logarithm and why is the exponential that you suggest of trace class ?
Come on man, now you're just trying to bust balls for no reason. Obviously one has to assume that the operators given are sufficiently nice so that the above formula makes sense. Which presumably is the case for interesting examples that arise in physics, in which case, if memory serves me, A is some nice second order differential operator on a compact manifold (like on a circle).

If you want to play the nit-pick game, you might as well object every time someone writes an equation like:

After all, what if x is zero, and which branch of log did you pick? Unless you want people to spend entire lifetimes composing on this forum, you're going to have to get used to a slightly looser style of exposition.
No you don't just assume that every operator in sight is "nice". That might work for some audiences, but it does not work in mathematics where one needs to be a bit more precise.

It may well be that there are lots of interesting examples that arise in physics wher the operators are sufficiently "nice". but that does not make for a complete and cogent theory. And that was the point that started this discussion, that the operations under discussion seem to work in physical situations for reasons that are not understood, and that they don't at this point have a firm mathematical foundation.

13. Originally Posted by DrRocket
No you don't just assume that every operator in sight is "nice". That might work for some audiences, but it does not work in mathematics where one needs to be a bit more precise.

It may well be that there are lots of interesting examples that arise in physics wher the operators are sufficiently "nice". but that does not make for a complete and cogent theory. And that was the point that started this discussion, that the operations under discussion seem to work in physical situations for reasons that are not understood, and that they don't at this point have a firm mathematical foundation.
I'm a bit confused by your objection here. Can I not just declare that we'll only attempt to define this sort of thing for squares of Dirac operators on compact manifolds (or some other general class of operators for which the definition makes sense)? What is the problem?

Incidently, I have a Ph.D. in pure math, and currently hold a post-doctoral research position. So I know what "works" as far as mathematical explanations go.

I admit that I am not an expert on zeta function regularization. All I was attempting to explain however was that the procedure is well-defined for a rather general class of differential operators.

Of course one can go on to ask more interesting questions such as (1) what does this procedure correspond to "physically" and (2) why does it contribute to the production of interesting mathematical invariants? I don't know the answer to either question, but I wouldn't be quick to conclude that the answer isn't known.

14. Originally Posted by salsaonline
Originally Posted by DrRocket
No you don't just assume that every operator in sight is "nice". That might work for some audiences, but it does not work in mathematics where one needs to be a bit more precise.

It may well be that there are lots of interesting examples that arise in physics wher the operators are sufficiently "nice". but that does not make for a complete and cogent theory. And that was the point that started this discussion, that the operations under discussion seem to work in physical situations for reasons that are not understood, and that they don't at this point have a firm mathematical foundation.
I'm a bit confused by your objection here. Can I not just declare that we'll only attempt to define this sort of thing for squares of Dirac operators on compact manifolds (or some other general class of operators for which the definition makes sense)? What is the problem?

Incidently, I have a Ph.D. in pure math, and currently hold a post-doctoral research position. So I know what "works" as far as mathematical explanations go.

I admit that I am not an expert on zeta function regularization. All I was attempting to explain however was that the procedure is well-defined for a rather general class of differential operators.

Of course one can go on to ask more interesting questions such as (1) what does this procedure correspond to "physically" and (2) why does it contribute to the production of interesting mathematical invariants? I don't know the answer to either question, but I wouldn't be quick to conclude that the answer isn't known.
I don't doubt that you can define the process for a large and interesting class of operators.

But we started this with my contention that the general process of renormalization in quantum field theory remains one that is not clearly defined in general. It is at best ad hoc and contains arbitrary steps.

I have no problem with your declaring that you will only use the procedure for a specific class of operators. In fact that seems to me to be necessary. By doing that you will have clearly defined the problem and introduced hypotheses that should permit one to understand the operations that you are describing. That is a perfectly normal way to go in pure mathematics.

I have no problem with pure mathematics. None at all. If more attention were paid to the kind of rigor that one finds in that discipline a lot of misconceptions and miscommunication would be avoided.

BTW a Ph.D. in pure math and a post-doc don't necessarily guarantee anything, but they sure go a long way towards credibility and I see no particular reason to doubt your capabilities. Most of the pure mathematicians of my acquaintance (and there are a lot of them) are pretty competent, and the exceptions are few and far between. I wish I could say the same of some other disciplines.

15. As I noted earlier, I believe that regularization and renormalization are two different things--that is, different ways of dealing with infinities that arise in different places.

The sense I get from reading things like the Clay book "Mirror Symmetry", or A. Zee's "Quantum Field Theory in a Nutshell" is that at least a few renormalization procedures are reasonably well-understood, and have a genuine physical interpretation.

16. Originally Posted by salsaonline
Originally Posted by Arcane_Mathamatition
okay, this all is very confusing to me, but I must ask, what in the name of all that is holy is ?
I assume you are writing for sqrt(-1). Anyway,

pretty much by definition.
Correct me if I'm wrong, but shouldn't that be , or is that what you mean (i.e. )

17. Originally Posted by thyristor
Originally Posted by salsaonline
Originally Posted by Arcane_Mathamatition
okay, this all is very confusing to me, but I must ask, what in the name of all that is holy is ?
I assume you are writing for sqrt(-1). Anyway,

pretty much by definition.
Correct me if I'm wrong, but shouldn't that be , or is that what you mean (i.e. )
In pure mathematics circles, by default refers to the logarithm base .

18. Ah, I see. Well, I'm sorry i made that post then. :-D

19. Originally Posted by thyristor
Originally Posted by salsaonline
Originally Posted by Arcane_Mathamatition
okay, this all is very confusing to me, but I must ask, what in the name of all that is holy is ?
I assume you are writing for sqrt(-1). Anyway,

pretty much by definition.
Correct me if I'm wrong, but shouldn't that be , or is that what you mean (i.e. )
In most contexts in mathematics log without any further comment is taken to mean the natural logarithm or ln.

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