Analytic continuation and physics

In summary, analytic continuation can be used in mathematics to assign a finite value to an infinite series that diverges to infinity. Is it correct and legitimate to equate this value to a diverging infinite series that occurs in a physical theory of nature? Will this process give a correct answer that can be verified by a physics lab experiment?
  • #1
kochanskij
44
4
Analytic continuation can be used in mathematics to assign a finite value to an infinite series that diverges to infinity. Is it correct and legitimate to equate this value to a diverging infinite series that occurs in a physical theory of nature? Will this process give a correct answer that can be verified by a physics lab experiment?

A specific example:

The infinite series 1+2+3+4+5+... occurs in a calculation for the Casimir Effect. This series is the Zeta Function at s = -1. Z(s) = 1/1^s + 1/2^s + 1/3^s + ... The analytic continuation of Z(-1) is -1/12 Is it correct to say that 1+2+3+4+5+... = -1/12 ? If you use the value -1/12 in your calculation of the Casimir force acting on parallel conducting plates and then measure this force in the lab, will your calculation agree with experiment?

Another example:

The Zeta function at s = 0 yields the series 1+1+1+1+1+... The analytic continuation of Z(0) is -1/2 This series occurs in String Theory when calculating the number of space-time dimensions. Is it correct to say that 1+1+1+1+1+... = -1/2 and use this value in your calculation? The result comes out to be 10 dimensions. Should we expect nature to agree with this calculation?
 
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  • #2
If the physical theory (like in the cases you described) can be modeled with the Zeta function, then it's OK.
 
  • #3
In the physical theories that I refer to, the Zeta function does NOT occur. Just the infinite series 1+2+3+4+... or the infinite series 1+1+1+1+... occurs. My question is: Is it correct to equate the finite values of -1/12 and -1/2 to these series?
It seems absurd to claim that the sum of an infinite amount of positive integers can equal a negative fraction! Can anyone justify this claim? (I understand analytic continuation, but I still don't think it applies to these diverging series.

A contradiction: In the first series, replace 2 with 1+1, 3 with 1+1+1, 4 with 1+1+1+1, and so on. We then have the sum of an infinite number of 1's. So these two series should sum to the same value. But analytic continuation gives -1/12 for the first series and -1/2 for the second. Can this contradiction be explained?
 
  • #4
The formula ## \zeta(0)=-\frac{1}{2}## is proved here

http://planetmath.org/valueoftheriemannzetafunctionats0

using the functional equation for ## \zeta ## and the formula for ## \zeta ## in the critical strip, the second ##1+ 2+3+4+ ... =-\frac{1}{12}## is the famous Ramanujan sum, you can see a proof here

http://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯

the criterion to obtain these kinds of series is different from the classic criterion of convergence , this is a sort of ''mean convergence'' (it is not a contradiction, it is another way to interpret a sum ...), as for exampe

[tex] 1-1+1-1+...=\frac{1}{2}[/tex]

if you do the average of the partial sums you have ## \frac{1+0}{2}=\frac{1}{2}, \frac{1+0+1}{3}=\frac{2}{3}, \frac{1+0+1+0}{4}=\frac{1}{2}, ... ## and these converge to ## \frac{1}{2}##

you can use the analytic continuation to extend ##\zeta## to this value.

Hi,
Ssnow
 

1. What is analytic continuation and how is it used in physics?

Analytic continuation is a mathematical technique used to extend a function from its domain of definition to a larger domain, where the function may not be defined at some points. In physics, it is used to extend physical laws and theories from their known domains to new domains, allowing for the prediction of new phenomena and the understanding of complex physical systems.

2. How does analytic continuation relate to complex numbers?

Analytic continuation is based on the properties of complex numbers, as it involves extending a function to the complex plane and using complex analysis techniques to determine its behavior. Complex numbers are crucial in physics, as many physical quantities, such as electric and magnetic fields, can be described using complex numbers.

3. Can analytic continuation be used to solve physical problems?

Yes, analytic continuation is an important tool in solving physical problems. It allows for the extension of physical theories to new domains, which can lead to the discovery of new phenomena and provide a deeper understanding of complex physical systems. It is also used in the study of quantum field theory and in the description of black holes in general relativity.

4. What are some examples of physical applications of analytic continuation?

One example is the extension of the Riemann zeta function to the entire complex plane, which has applications in number theory and the study of prime numbers. Another example is the use of analytic continuation in the calculation of scattering amplitudes in quantum field theory, which is essential in understanding the behavior of subatomic particles.

5. Are there any limitations to using analytic continuation in physics?

While analytic continuation is a powerful tool in physics, it is not always applicable. In some cases, the extension of a function using analytic continuation may lead to incorrect results or may not be physically meaningful. Additionally, the use of analytic continuation in physics often requires making certain assumptions and approximations, which may limit its accuracy in certain situations.

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