Wikipedia:Reference desk/Archives/Mathematics/2016 September 13

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September 13

What is the zero root of 10

Cube root of 10 is 2.1544

Square root of 10 is 3.1622

My question is

What is the Unity root of 10?

What is the Zero root of 10?

Ohanian (talk) 16:25, 13 September 2016 (UTC)

Why stop there?

What is the negative unity root of 10? x^(-1)=10

What is the negative square root of 10? x^(-2)=10

What is the negative cube root of 10? x^(-3)=10

Ohanian (talk) 16:31, 13 September 2016 (UTC)

For every x, the 1-st root (the "unity-root" as you call it) of x is x. No number has a zero-th root. For every x,n (other than zero), the minus-n-th root of x, is one divided by the n-th root of x. HOTmag (talk) 17:22, 13 September 2016 (UTC)

To explain why, the root of a number is the number raised to the inverse of the power. That is, the square root of x is x^1/2, or more generally, the "y" root of "x" is x^1/y. The zeroth root would thus be undefined, because the zeroth root would be x^1/0, 1/0 is undefined. See division by zero. --Jayron32 17:52, 13 September 2016 (UTC)

The zeroth root of any number greater than 1 is undefined for the same reason any nonzero number divided by 0 is undefined. The zeroth root of 1 is undefined because it is an indeterminate expression, closely related to the form 1 to the infinity that's included in the indeterminate form article. (I believe the reason the zeroth root of 1 isn't included as a separate expression is because roots are usually not preferred in calculus; fractional exponents are preferred, in which case it becomes 1^(1/0), which is just a variant of the included 1 to the infinity.) I find it very natural to think that the zeroth root of any number 0 <= n < 1 is 0 because defining it as such is consistent with how limits usually work; they're not indeterminate and the limit is a real number, not +-infinity. But nobody really defines these zeroth roots as such. Any reason?? Georgia guy (talk) 17:56, 13 September 2016 (UTC)

I think "roots" in general are an algebraic concept more than an analytic one. Saying ${\sqrt[{n}]{x}}=y$ means that if you multiply together n copies of y, you get x. The n is specifically a natural number, not a real number. You can't take "limits" over the natural numbers; they're discrete.

Certainly, if you like, you can define ${\sqrt[{y}]{x}}=x^{\frac {1}{y}}$ for x and y both real. It's a natural enough notation; no one will say it's "wrong". But it isn't used very much.

As for your point that the limit of $x^{\frac {1}{y}}$ is zero as y goes to zero, if x is between 0 and 1 — not really. That's only if you use the one-sided limit, from the right. As y goes to zero from the left, the expression goes to infinity. --Trovatore (talk) 18:34, 13 September 2016 (UTC)

As for your comment about real exponents and about the impossibility of taking limits over the naturals. Well, indeed, by using the initial definitions $x^{1}=x,x^{a+1}=x^{a}x$, the formula $x^{a+b}=x^{a}x^{b}$ is only provable for every natural $a,b$. However, when the latter formula - is extended to the domain of every rational $a,b$ - and is used along with the initial definition $a^{1}=a$, then one gets the (real non-negative) value of $x^{a}$ for every rational $a$ (with a real non-negative $x$). Hence, by taking limits over the rationals, one gets also the (real non-negative) value of $x^{a}$ for every real $a$ (with a real non-negative $x$).

As for ${\sqrt[{y}]{z}}=x$: Yes, by the initial definition: ${\sqrt[{y}]{z}}=x$ iff $x^{y}=z$, one gets the (real non-negative) value of ${\sqrt[{y}]{z}}$ (with real positive $y,z$, excluding the undefined value of ${\sqrt[{0}]{1}}$) - even without using the "artificial" formula ${\sqrt[{y}]{z}}=z^{\frac {1}{y}}$. However, this artificial formula is not a "definition" but rather is a consequence (obtained from the above initial definitions). HOTmag (talk) 23:18, 13 September 2016 (UTC)

Well, as soon as you take limits, you're out of the realm of algebra and into analysis. My claim is that "roots" are really a concept from algebra, not analysis.

As for a concept of the q'th root of x, where q is rational — sure, again, you can do it; it's pretty clear what it means (and it's arguably still algebra). But why do you want to? In practice, hardly anyone does. --Trovatore (talk) 01:19, 14 September 2016 (UTC)

If the limits from both directions agreed, I would be tempted to take that as the answer. However, since the curve is discontinuous at 0, that supports the idea that it's undefined. StuRat (talk) 01:42, 14 September 2016 (UTC)

For proving that no number has a zeroth root, you don't need the discontinuity at zero. You only need the definition: ${\sqrt[{y}]{z}}=x$ if and only if $x^{y}=z$. HOTmag (talk) 08:31, 14 September 2016 (UTC)

It seems like we agree about all.

I've only referred to your argument that limits cannot be taken over "natural" exponents, so I pointed out that you can still take limits over "rational" exponents. So, instead of your saying that limits cannot be taken over "natural" numbers, it would suffice to point out that taking limits at all (whether over natural numbers or over rational numbers) is outside the realm of algebra.

As for your comment about a concept of the $q$-th root of $z$: No, I didn't refer to a necessarily non-natural $q$. I've only referred to your claim that one can "define" ${\sqrt[{q}]{z}}=z^{\frac {1}{q}}$ (for $q$ and $z$ both real [ $q$ being other than zero ] ), so I pointed out, that's not a "definition" (whether $q$ is natural or not), but rather is a consequence - obtained from the combination of three initial definitions: ${\sqrt[{q}]{z}}=x$ iff $x^{q}=z,$ and $x^{a+b}=x^{a}x^{b},$ and $x^{1}=x.$ HOTmag (talk) 08:31, 14 September 2016 (UTC)

Functionals, functional derivatives, functional integration and functional equations

Text between solid lines is moved here from the math project where I accidentally posted first.

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Is there a rigorous treatment of these subjects as used in quantum field theory? In particular, I am looking for a good reference text at the graduate level.

In QFT functionals are often operator valued (even the Lagrangian may be treated this way, implicitly or (rarely) explicitely), presumably they evaluate to operators on a Hilbert space (but the Hilbert space may be yet to be defined); for instance, there are quantum mechanical functional derivatives around that are operators. Naturally, no physicist with self esteem would ever go past mention (usually not even that) that we are (sometimes, not always!) dealing with operator valued thingies, and then they proceed in a purely formal manner. I'd like to see the concepts in the header somewhat rigorously defined, including when operator valued functionals are involved. Thanks in advance. YohanN7 (talk) 11:12, 13 September 2016 (UTC)

Do you mean linear functionals? Or even (still more specifically) something like operator-valued generalized functions? Boris Tsirelson (talk) 12:35, 13 September 2016 (UTC)

(EC. I tweaked my OP a bit.). No, they don't necessarily have to be linear. Take e.g. the Lagrangian for a scalar field theory. It can be interpreted in the ordinary sense as well as operator valued. I'll have a look at the linked article and get back. Thanks. YohanN7 (talk) 12:41, 13 September 2016 (UTC)

An explicit example: One may define a quantum mechanical functional derivative by

${\frac {\delta F[q(t),p(t)]}{\delta q(\mathbf {x} ,t)}}=[p(\mathbf {x} ,t),F[q(t),p(t)]],$

and similarly for p provided the p, q have the "canonical commutation relations" of QM. Disclaimer: A factor of i and possibly a minus sign is missing. In addition, anticommutator relations are allowed under some circumstances. This does make sense provided that the ordering of the p, q in F is according to a certain prescription.YohanN7 (talk) 12:49, 13 September 2016 (UTC)

It is this sort of things I'd like a good reference to. YohanN7 (talk) 12:53, 13 September 2016 (UTC)

Example from around here: Scalar field theory#Quantum scalar field theory. The Hamiltonian given there is of the sort I am referring to. It could equally well be a quantum representation (operator) of a Lie algebra element of a corresponding symmetry group. These things are not mysteries to me, but I'm still looking for a text, preferably written by a mathematician, treating the mathematical details that are undoubtedly there, but swept under the carpet. YohanN7 (talk) 14:04, 13 September 2016 (UTC)

I doubt that such paradise exists... Maybe try Chapter 7 "Quantum fields" of "Quantum mechanics and the particles of nature: An outline for mathematicians" by A. Sudbery (Cambridge Univ. Press, 1986). Also (maybe) "White Noise Calculus and Fock Space" by N. Obata; and "White Noise: An Infinite Dimensional Calculus" by T. Hida, H-H. Kuo, J. Potthoff, L. Streit. Boris Tsirelson (talk) 15:01, 13 September 2016 (UTC)

But you know, angel with the flaming sword protects the paradise (of QFT) from mathematicians; his sword is renormalization. Boris Tsirelson (talk) 16:40, 13 September 2016 (UTC)

Boris, you are a poet! There have been attempts at treatment of renormalization from more mathematical points of view, like Costello's book Renormalization and Effective Field Theory, but yes, I've seen nothing rigorous. --Mark viking (talk) 18:05, 13 September 2016 (UTC)

An interesting question. There are formal approaches such as axiomatic QFT and algebraic QFT that can take the operator POV, but they have not been extended successfully to realistic QFTs. The calculus of variations is based on an algebra of commutative functions, so the mathematical approaches I have seen--geometric quantization, topological QFTs--all drop down from the Heisenberg picture to a representation with commutative functions in which one can talk about manifolds, symplectic forms, etc., and actually compute. If there is a rigorous operator-based calculus of variations that can deal with the fact that the time-ordered product $T[\phi (x)\phi (x^{\prime })]$ in a typical QFT blows up as $x\to x^{\prime }$ (Boris' archangel), I have not seen it. That said, the more narrow problem of time-ordering in a perturbative expansion has been pretty thoroughly considered by physicists, if not put on a rigorous foundation by mathematicians. See for instance papers on Dyson-Wick expansions like [1]. --Mark viking (talk) 18:05, 13 September 2016 (UTC)

Yes... and Feynman integral suggests that we should mind a complex-valued "measure" on a space of functions; the "measure" is of infinite variation, far beyond measure theory; and the relevant class of functions should depend on the dynamics (Hamiltonian or Lagrangian); Fock space is good only for free fields. "In fact, renormalization is the major obstruction to making path integrals well-defined." (quoted from "Path integral formulation#The need for regulators and renormalization"). Boris Tsirelson (talk) 05:35, 14 September 2016 (UTC)

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Thank you for interesting replies and references – and for the poetry. There is an old maxim that says "shut up and compute" when someone wonders about the interpretations of quantum mechanics. That is perhaps good advice here as well. Anyway, I'll hit the linked material, though it will undoubtedly cause pain. But i enjoy this particular kind of pain. YohanN7 (talk) 07:27, 16 September 2016 (UTC)