## A Notable Quote on Richard Feynman

From Freeman Dyson, “Disturbing the Universe”, Harper & Row, New York, 1979:

“Dick was also a profoundly original scientist. He refused to take anybody’s word for anything. This meant that he was forced to rediscover or reinvent for himself almost the whole of physics. It took him five years of concentrated work to reinvent quantum mechanics. He said that he couldn’t understand the official version of quantum mechanics that was taught in textbooks, and so he had to begin afresh from the beginning. This was a heroic enterprise. He worked harder during those years than anybody else I ever knew. At the end he had a version of quantum mechanics that he could understand.”

This quote is, of course, about Richard Feynman‘s path integral formulation of quantum mechanics.

## $\sum_{n=0}^\infty e^{nix}$ is Cesàro Summable

Back when I was a Ph.D. student, a friend of mine (he was a Ph.D. student in physics studying laser optics) asked me if the series $\sum_{n=0}^\infty e^{nix}$ converges. I vaguely remember that his advisor needed to have a finite value for the infinite sum for whatever reason I don’t remember. At that time, I didn’t know other summability methods and I only knew the conventional definition of infinite sums. The series is a geometric series with $r=e^{ix}$ and since $|e^{ix}|=1$, I bluntly told him that you can’t have a finite value for the series. It diverges!

I don’t know what motivated me but I just thought about the series $\sum_{n=0}^\infty e^{nix}$ and I was wondering if it is Cesàro summable. I was able to show that indeed it is. To my pleasant surprise, the Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ is $\frac{1}{1-e^{ix}}$. I am sure that this is well-known and I am just being ignorant about it. I would appreciate if someone can tell me any reference where the Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ appears.

The $(n+1)$-th partial sum $S_{n+1}$ of $\sum_{n=0}^\infty e^{nix}$ is
\begin{align*}
S_{n+1}&=\sum_{k=0}^n e^{nix}\\
&=1+e^{ix}+e^{2ix}+\cdots+e^{nix}\\
&=\frac{1-e^{(n+1)ix}}{1-e^{ix}}
\end{align*}
provided $x\ne 2m\pi$, $m\in\mathbb{Z}$. Now the sum of the first $n$ partial sums is calculated to be
$$\sum_{k=0}^{n-1}S_k=\frac{1}{1-e^{ix}}\left[(n+1)-\frac{1-e^{(n+1)ix}}{1-e^{ix}}\right].$$
The Cesàro sum of $\sum_{n=0}^\infty e^{nix}$ is
\begin{align*}
\sum_{n=0}^\infty e^{nix}&=\lim_{n\to\infty}\frac{\sum_{k=0}^{n-1}S_k}{n}\\
&=\frac{1}{1-e^{ix}}
\end{align*}
as $\lim_{n\to\infty}\frac{1}{n}\frac{1-e^{(n+1)ix}}{1-e^{ix}}=0$. For $x=2m\pi$ with $m\in\mathbb{Z}$, $\sum_{n=0}^\infty e^{nix}=1+1+1+\cdots$ is not Cesàro summable as its Cesàro sum is \begin{align*}
1+1+1+\cdots&=\lim_{n\to\infty}\frac{\sum_{k=1}^n S_k}{n}\\
&=\lim_{n\to\infty}\frac{\frac{n(n+1)}{2}}{n}\\
&=\infty.
\end{align*}
However, it can be shown that $1+1+1+\cdots=-\frac{1}{2}$ using zeta function regularization. (I also discussed it here.)

The real part of Cesaro sum with n=300

This figure shows the real part of the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to the real part of $\frac{1}{1-e^{ix}}=\frac{1}{2}$.

The imaginary part of Cesaro sum with n=300

This figure shows the imaginary part of the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to the imaginary part of $\frac{1}{1-e^{ix}}=\frac{\sin x}{2(1-\cos x)}$.

Cesaro sum with n=100

Finally this figure shows the Cesàro sum $\sum_{n=0}^\infty e^{nix}$ with $x=\frac{\pi}{6}$ converging to $\frac{1}{1-e^{ix}}$.

Posted in Summability Methods | 4 Comments

## $1+2+3+4+\cdots=-\frac{1}{12}$?

No, folks! I am not drunk nor I am pot-headed. Yet, I am about to discuss the crazy identity
$$1+2+3+4+\cdots=-\frac{1}{12}.$$
No, I am not joking either. This is actually pretty serious mathematics and is also pretty serious stuff even to physicists. I promise you that by the time you finish reading this blog article, it will all make sense to you (I hope). So, please bear with me.

The very first thing we learned about numbers in elementary school was how to add two numbers, and that was not bad. But when it came to adding three numbers, things were confusing. Because we didn’t know what $1+2+3$ meant. It could mean $(1+2)+3$ i.e. add 1 and 2 first, and then add the resulting number (which is 3) to 3. Or, it could mean $1+(2+3)$ i.e. add 2 and 3 first and then add 1 to the resulting number (which is 5). It turns out that whichever you do it does not matter. They will all turn out to be the same number 6. In fact, for any real numbers $a$, $b$ and $c$, the following property holds:
$$(a+b)+c=a+(b+c).$$
This property, as we recall, is called the associative law. By the induction process, we know that what is true for three numbers is true for any finitely many numbers. For instance, knowing that the associative law holds for three numbers, we also get
$$(a+b+c)+d=a+(b+c+d)$$
for any real numbers $a,b,c,d$. This means that the sum $a+b+c+d$  can be obtained by any of the following four ways
$$[(a+b)+c]+d,\ [a+(b+c)]+d,\ a+[(b+c)+d],\ a+[b+(c+d)].$$
For multiplication, while complex numbers (2-dimensional numbers) and quaternions (4-dimensional numbers) satisfy the associative law, octonions (8-dimensional numbers) do not. The associative law, on the other hand, does not hold in general when we add infinitely many numbers. For instance, let us consider the Grandi’s series
$$1+(-1)+1+(-1)+\cdots.$$
If we assume that the associative law still holds for this case, we can prove something interesting. First, by the associative law we obtain
\begin{align*}
1+(-1)+1+(-1)+\cdots&=(1-1)+(1-1)+(1-1)+(1-1)+\cdots\\
&=0+0+0+0+\cdots\\
&=0.
\end{align*}
But then again by the associative law we also obtain
\begin{align*}
1+(-1)+1+(-1)+\cdots&=1+(-1+1)+(-1+1)+(-1+1)+\cdots\\
&=1+0+0+0+\cdots\\
&=1.
\end{align*}
Therefore, we just proved that $0=1$. Of course this is bullshit! Now that we know the associative law does not hold in general for infinite sums, our question is what do we mean by adding an infinitely many numbers? That is what do we mean by
$$\sum_{k=1}^\infty a_k=a_1+a_2+a_3+\cdots+a_k+\cdots?$$
Although we think that we can perceive the notion of infinity, our brain can actually process only finitely many things just like computers do. So this is the way we perceive natural numbers. We don’t actually perceive the entire infinitely many natural numbers. We can only count finitely many of them but our mind can convince us that the process continues indefinitely as 2 comes after 1, 3 comes after 2,$\cdots$,  1 million 1 comes after 1 million, and so on so forth. This is the way we perceive infinity. We do not perceive actual infinity but only potential infinity through finite processes. From intuitionism point of view, the actual infinity such as the set $\mathbb{N}$ of all natural numbers is an illusion and it should not be considered as a mathematical object. Finitism even rejects the notion of potential infinity and says that “a mathematical object does not exist unless it can be constructed from the natural numbers in a finite number of steps.” (By the way I am not an intuitionist but a
Platonist.) Back to our previous question. What we really can do is the finite sum
$$s_n=\sum_{k=1}^n a_k=a_1+a_2+a_3+\cdots+a_n,$$
which is called the $n$-th partial sum, but we can define the infinite sum $\displaystyle\sum_{k=1}^\infty a_k$ as the limit of the $n$-th partial sum $s_n$,  $\displaystyle\lim_{n\to\infty}s_n$. If this limit exists as a finite number, we say the infinite sum $\displaystyle\sum_{k=1}^\infty a_k$ exists. If the limit does not exist or becomes $\infty$ or $-\infty$, we say the infinite sum does not exist. The $n$-th partial sum $s_n$ for the Grandi’s series is $1,0,1,0,1,0,\cdots$ so the sequence of partial sums $\{s_n\}$ does not converge, and hence the Grandi’s series does not converge in ordinary sense i.e. the way we learned in calculus. Although this definition of infinite sums appears to be most natural and intuitive, there may be other legitimate ways to define infinite sums. In fact, there are. One of them is Cesàro’s sum. Cesàro’s sum of an infinite series is defined by
$$\lim_{n\to\infty}\frac{\displaystyle\sum_{k=1}^ns_k}{n}$$
i.e. the limit of the arithmetic mean of the first $n$ partial sums as $n\to \infty$. If an infinite series is summable, it is Cesàro summable. But the converse need not be true. A counterexample for the converse is Grandi’s series. The partial sums of Grandi’s series are
$$\frac{1}{1},\frac{1}{2},\frac{2}{3},\frac{2}{4},\frac{3}{5},\frac{3}{6},\frac{4}{7},\frac{4}{8},\cdots$$
and the limit of this sequence is $\frac{1}{2}$ as seen in the following picture.

Cesaro’s sum of Grandi’s series

There is another summation method called Abel summation which is more powerful than Cesàro’s summation. It uses a different mean called the abelian mean. Let $\{\lambda_n\}_{n=0}^\infty$ be a strictly increasing sequence such that $\lambda_0\geq 0$ and $\displaystyle\lim_{n\to\infty}\lambda_n=\infty$. Let $f(x)=\displaystyle\sum_{n=0}^\infty a_ne^{-\lambda_nx}$. Suppose that $f(x)$ converges for all real numbers $x>0$. Then the abelian mean $A_\lambda$ is defined as
$$A_\lambda(s)=\lim_{x\to 0+}f(x).$$
Now let $\lambda_n=n$. Then we obtain
$$f(x)=\sum_{n=0}^\infty a_ne^{-nx}=\sum_{n=0}^\infty a_nz^n\ (z=e^{-x})$$
and
$$\lim_{x\to 0+}f(x)=\lim_{z\to 1-}\sum_{n=0}^\infty a_nz^n.$$
This limit is called Abel summation. Let us consider the infinite sequence
$$\{a_n\}=\{(-1)^n(n+1):n=0,1,2,\cdots\}=\{1,-2,3,-4,\cdots\}.$$
The series $\displaystyle\sum_{n=0}^\infty a_n$ is not summable nor is Cesàro summable as seen in the following pictures.

Cesaro’s sum of 1-2+3-4+…

But it is Abel summable. To see this,
\begin{align*}
\sum_{n=0}^\infty a_nz^n&=1-2z+3z^2-4z^3+\cdots\\
&=\frac{1}{(1+z)^2}
\end{align*}
for $|z|<1$. Thus,
$$\lim_{z\to 1-}\sum_{n=0}^\infty(-1)^n(n+1)z^n=\lim_{z\to 1-}\frac{1}{(1+z)^2}=\frac{1}{4}.$$
This means than
$$1-2+3-4+\cdots=\frac{1}{4}$$
as Abel summation. Abel summation also can calculate Grandi’s series. Let $a_n=(-1)^n$, $n=0,1,2,\cdots$. Then
\begin{align*}
\sum_{n=0}^\infty a_nz^n&=1-z+z^2-z^3+\cdots\\
&=\frac{1}{1+z}
\end{align*}
for $|z|<1$. Hence, we obtain
$$\lim_{z\to 1-}\sum_{n=0}^\infty a_nz^n=\lim_{z\to 1-}\frac{1}{1+z}=\frac{1}{2}$$
as we have seen earlier. From this we see that
$$(1-1+1-1+\cdots)^2=1-2+3-4+\cdots.$$
This identity can be also obtained by the Cauchy product. Cesàro summation or Abel summation can be used to calculate oscillating divergent series to possibly produce a finite answer. However, they cannot produce a finite answer for a series that diverges to $\infty$. For example, $1+2+3+4+\cdots$ is neither Cesàro summable nor Abel summable.

Interestingly Srinivasa Ramanujan (1877-1920) showed in his notebook that
$$1+2+3+4+\cdots=-\frac{1}{12}$$
and his proof is pretty elementary. Let $c=1+2+3+4+\cdots$. Then
\begin{aligned} c&= 1&+&2+3&+&4+5&+&6&+&\cdots\\ 4c&= &+&4 &+&8 &+&12&+&\cdots \end{aligned}
Subtracting the second identity from the first results
$$-3c=1-2+3-4+5-6+\cdots=\frac{1}{4}$$
and hence we obtain
$$c=-\frac{1}{12}\ \mbox{i.e.}\ 1+2+3+4+\cdots=-\frac{1}{12}.$$
This appears to be a way too elementary and simple proof for the outrageous claim. The problem with this proof is that it is carried out by assuming that $c$ is a finite number. But we don’t know that, do we? In ordinary sense, $c=\infty$ so subtracting $4c$ from $c$ would result $\infty-\infty$ on the left hand side which is undefined, while it would result $1-2+3-4+5-6+\cdots$ on the right hand side. You need to be very careful when you deal with infinite sums and treating them like finite numbers is dangerous as it may result an inconsistent result. While I am not happy with Ramanujan’s proof, what he claimed may still be true. And yes, I am still sober. In fact, there is a much more sophisticated way to show that $1+2+3+4+\cdots=-\frac{1}{12}$. It is called zeta function regularization. Let
$$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$$
where $s=\sigma+it$ is a complex variable. $\zeta(s)$ converges for all complex numbers $s$ with $\sigma=\mathrm{Re}(s)>1$. $\zeta(s)$ diverges when $\sigma=\mathrm{Re}(s)\leq 1$, in particular when $s=-1$ in which case we obtain
$$\zeta(-1)=1+2+3+4+\cdots.$$
Bernhard Riemann (1826-1866) showed that $\zeta(s)$ can be continued analytically to the punctured plane $\mathbb{C}\setminus\{1\}$. The analytic continuation of $\zeta(s)$ is called the Riemann zeta function. If you are not familiar with analytic continuation, I explained the idea of analytic continuation using a simple example here. The Reimann zeta function is a meromorphic function on $\mathbb{C}$, which is holomorphic everywhere except for a simple pole at $s=1$.
\begin{aligned} \zeta(s)&=1^{-s}&+&2^{-s}&+&3^{-s}&+&4^{-s}&+&5^{-s}&+&6^{-s}&+&\cdots\\ 2\cdot 2^{-s}\zeta(s)&=0&+&2\cdot 2^{-s}&+&0&+&2\cdot 4^{-s}&+&0&+&2\cdot 6^{-s}&+&\cdots \end{aligned}
Subtracting the second identity from the first results
$$(1-2^{1-s})\zeta(s)=1^{-s}-2^{-s}+3^{-s}-4^{-s}+5^{-s}-6^{-s}+\cdots=\eta(s).$$
The Dirichlet series
$$\eta(s)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+1)^s}=\frac{1}{1^s}-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+\cdots$$
converges only for complex numbers $s$ with $\mathrm{Re}(s)>0$. However, $\eta(s)$ is Abel summable for any complex number $s$. Hence, it can be analytically continued to the entire complex plane $\mathbb{C}$. The analytic continuation of $\eta(s)$ which is an entire function is called the Dirichlet eta function. The identity $(1-2^{1-s})\zeta(s)=\eta(s)$ still holds when both functions are continued analytically to the punctured plane $\mathbb{C}\setminus\{1\}$. Substituting $s=-1$, we obtain $-3\zeta(-1)=\eta(-1)$. $\eta(-1)=1-2+3-4+5-6+\cdots$ diverges to $\infty$ but it is Abel summable and the Abel sum of the series is $\frac{1}{4}$
$$\eta(-1)=\lim_{z\to 1-}(1-2z+3z^2-4z^3+\cdots)=\lim_{z\to 1-}\frac{1}{(1+z)^2}=\frac{1}{4}$$
as we have seen earlier. $\eta(-1)$ as the analytic continuation of $\eta(s)$ evaluated at $s=-1$ is the Abel sum $\frac{1}{4}$. Therefore, we obtain $\zeta(-1)=-\frac{1}{12}$ i.e. $1+2+3+4+\cdots=-\frac{1}{12}$. So, have we actually proved $1+2+3+4+\cdots=-\frac{1}{12}$ now? The answer is yes and no. No, no, I am not playing with you. Let me explain. What we actually have proved here is $\zeta(-1)=-\frac{1}{12}$ and in fact you also remember that $\zeta(-1)$ is not defined because it diverges. So here $\zeta(-1)$ is not really $1+2+3+4+\cdots$ but the Riemann zeta function i.e. the analytic continuation of $\zeta(s)$ evaluated at $s=-1$. Precisely speaking, it is not that $1+2+3+4+\cdots=-\frac{1}{12}$ but that we can assign the infinite series $1+2+3+4+\cdots$ a unique finite number $-\frac{1}{12}$, which coincides with Ramanujan’s calculation, using the analytic continuation of $\zeta(s)$. As an analogy, in my notes here, $\displaystyle f_1(s)=\sum_{n=0}^\infty(-1)^ns^n$ is not defined at $s=\frac{3}{2}i$ because $f_1\left(\frac{3}{2}i\right)$ diverges to $\infty$, however the analytic continuation $F(s)$ is defined at $s=\frac{3}{2}i$ and that $F\left(\frac{3}{2}i\right)=\frac{4}{13}-\frac{6}{13}i$. Hence, we may assign the divergent series $f_1\left(\frac{3}{2}i\right)$ a unique finite number $\frac{4}{13}-\frac{6}{13}i$.

Giving a finite value to a divergent quantity is not an unusual practice in mathematics. For instance, we can turn the infinite number line $\mathbb{R}=(-\infty,\infty)$ into a unit circle $S^1$ by assigning a finite point to the boundary $\{\pm\infty\}$ of $\mathbb{R}$. Here is one way to do it. The infinite number line $(-\infty,\infty)$ is homeomorphic to the open interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ via the map $f(t)=\arctan t$. This means that $f(t)$ is a homeomorphism i.e it is one-to-one and onto, is continuous and its inverse $f^{-1}(t)$ is also continuous ($f^{-1}(t)=\tan t$). In the eyes of topologists two homeomorphic spaces are the same i.e. they do not distinguish $(-\infty,\infty)$ and $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. The open interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ is then homeomorphic to $(0,2\pi)$ via the homeomorphism $g(s)=2s+\pi$. The open interval $(0,2\pi)$ is then homeomorphic to the unit circle $S^1$ with one point $(1,0)$ removed via the homeomorphism $h(\theta)=(\cos\theta,\sin\theta)$. The composition $(h\circ g\circ f)(t)$ is then a homeomorphism from $(-\infty,\infty)$ to $S^1\setminus\{(1,0)\}$. By adding a point $(1,0)$ to $S^1\setminus\{(1,0)\}$, we obtain a unit circle $S^1$. This process is called one-point compactification in topology. Similarly adding a point $\infty$ (this is called the ideal point) to the infinite plane $\mathbb{R}^2$ and identifying the boundary of $\mathbb{R}^2$ with the ideal point $\infty$ results a unit sphere $S^2$. This whole process can be done via the stereographic projection from the north pole of $S^2$ as seen in the following picture. In fact, the stereographic projection can be also used to show that we can obtain a unit circle $S^1$ by adding a point to the real line $\mathbb{R}$ (and this point is identified with the boundaries $\{\pm\infty\}$ of $\mathbb{R}$).

Stereographic projection from the north pole

In the study of instantons in physics, an infrared cut-off, the finite-energy condition imposed on Yang-Mills action with $|x|\to\infty$ where $x\in\mathbb{R}^4$, geometrically amounts to the one-point compactification of $\mathbb{R}^4$ which is the 4-sphere $S^4$.

In the strip $0<\mathrm{Re}(s)<1$, the zeta function satisfies the functional equation
$$\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s).$$
One can immediately see that $\zeta(s)=0$ for $s=-2,-4,-6,\cdots$, negative even integers. These are called trivial zeros of the Riemann zeta function and there are indeed nontrivial zeros of the Riemann zeta function as well. There is a famous conjecture, called the Riemann Hypothesis, regarding the nontrivial zeros of the Riemann zeta function, namely the nontrivial zeros $s$ of the Riemann zeta function all have real part $\mathrm{Re}(s)=\frac{1}{2}$. This conjecture still has not been resolved (proved or disproved). It is part of Hilbert’s eighth problem along with the Golbach conjecture in the Hilbert’s 23 unsolved problems and is also one of the Clay Mathematics Institute Millennium Prize Problems. Using the Riemman zeta function we can calculate a finite answer for another divergent series $1+1+1+\cdots$.
\begin{align*}
\zeta(0)&=\lim_{s\to 0}2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)\\
&=\frac{1}{\pi}\lim_{s\to 0}\sin\left(\frac{\pi s}{2}\right)\zeta(1-s)\\
&=\frac{1}{\pi}\lim_{s\to 0}\left(\frac{\pi s}{2}-\frac{\pi^3s^3}{48}+\cdots\right)\left(-\frac{1}{s}+\cdots\right)=-\frac{1}{2}.
\end{align*}
That is, we obtain
$$1+1+1+\cdots=-\frac{1}{2}.$$

This sort of regularization i.e. giving a finite value to a divergent quantity is particularly important to physicists along with renormalization as the divergence of path integrals often appears in quantum field theory even when physically you expect finite values for those integrals. Many physicists are trying to find the resolution for the divergences from mathematics such as regularization. However, I don’t think that the answer is in mathematics. I suspect that the occurrence of divergences in quantum field theory may have originated from its foundation and I believe that the answer can be found by carefully and thoroughly reexamining the way the current quantum field theory is formulated. For one, in quantum field theory particles are treated as mathematical points i.e. there are no inner structures of particles contrary to the nature of actual particles. Second, I also believe that the way the path integral was formulated is incorrect. (I am not saying that the idea of path integral formulation is wrong. I do believe that the idea itself is correct. I just believe that the use of complex numbers in the formulation of path integrals is incorrect. For this, see my blog article here.) I will delve into this issue at some other time when I have a better understanding.

Back to mathematics, so what now? Well, we have two conflicting types of arithmetic. One type of arithmetic, which coincides with our perception of numbers, says that $1+2+3+4+\cdots=\infty$ and the other type of arithmetic says that
$1+2+3+4+\cdots=-\frac{1}{12}$. While this second type of arithmetic appears to be mathematically consistent, it is also against everything we experience about numbers. For instance, we know that if we add any finitely many positive numbers, the result would still be positive. Now the infinite sum $1+2+3+4+\cdots$ is not only finite but also negative! The big question is how these two conflicting types of arithmetic can be consistent with each other and how do we cope with this trouble? The situation may be parallel to what happened in geometry about a couple centuries ago. The parallel postulate, also called Euclid’s fifth postulate (because it is the fifth postulate in Euclid’s Elements) states that, in two-dimensional geometry,

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.” In 1795, John Playfair (1748-1819) showed that Euclid’s parallel postulate can be replaced by the following axiom

At most one line can be drawn through any point not on a given line parallel to the given line in a plane.”

Many mathematicians thought Euclid’s parallel postulate might be proved from the other four postulates of Euclid and attempted to prove it and failed. Among them includes Adrien-Marie Legendre (1752-1833). In the beginning of 19th century, mathematicians realized the possibility of other geometries by negating the parallel postulate. Carl Friedrich Gauß (1777-1855) knew the possibility of non-Euclidean geometry but never publicized his finding, perhaps because he was afraid of criticisms from other mathematicians. There are two different cases with negating the Euclid’s parallel postulate. One is “In a plane, given a point $p$ and a line $\ell$ not passing through $p$, there exist two lines through $p$ which do not meet $\ell$.” Replacing the Euclid’s parallel postulate by this postulate, János Bolyai (1802-1860) and Nikolai Ivanovich Lobachevsky (1792-1856) independently discovered a non-Euclidean geometry, called hyperbolic geometry. In fact, in plane hyperbolic geometry given a point $p$ and a line $\ell$ not passing through $p$, there exist infinitely many lines through $p$ which do not meet $\ell$, i.e. there are infinitely many parallel lines. The other case is “In a plane, given a point $p$ and a line $\ell$ not passing through $p$, all the lines through $p$ meet $\ell$,” i.e. there are no parallel lines. Replacing the Euclid’s parallel postulate by this one, Bernhard Riemann discovered a non-Euclidean geometry called elliptic geometry which is the simplest case of Riemannian geometry. The existence of plane geometries with three different parallel postulates may appear to be a contradiction. Due to Arthur Cayley (1821-95), Eugenio Beltrami (1835-1912), Felix Klein (1849-1925), and Henri Poincaré (1854-1912), et. al. it turns out that non-Euclidean geometries can be modelled within Euclidean geometry, and hence there is no contradiction with having both Euclidean and non-Euclidean geometries and they can all be consistent with each other.

Having a lesson from geometry, one must wonder whether we are in the same situation with the two different types of arithmetic we have now, which appear to be conflicting with each other. We obtained $1+2+3+4+\cdots=-\frac{1}{12}$ within the conventional arithmetic through analytic continuation. To me this appears to be a reminiscence of obtaining models of non-Euclidean geometries within Euclidean geometry. But a suitable mathematical or physical interpretation of this new arithmetic still lacks. I cannot shake off the feeling that we are missing something big here with a chain of questions. What is the meaning of this new arithmetic? What are the implications of this new arithmetic regarding its possible impact on mathematics? Does it indicate that there may be a new type of mathematics we don’t yet know about? If so, what could be possible impact of such new mathematics on physics? I will get to these questions in due course and hopefully by then I will have answers for them.

## Time Travel, Parallel Universes (Alternate Realities) and the Greys

Time Travel in Science Fiction

It does not require a physicist to get excited about time travel. Time travel is one of the most popular themes in modern day science fiction genre. “The Time Machine, H. G. Wells, 1895″ is probably the first novel in which an instrument of time travel was described, although the notion of time travel appeared in many earlier stories such as a Japanese legend of a fisherman Urashima Taro or the story of Abimelech (the same person as Ebed-melech the Ethiopian of Jeremiah 38:7) in the pseudepigraphical text of the Old Testament. (It is part of the Ethiopian Orthodox Amharic Bible.) Examples of TV shows and Films where the notion of time travel appeared include: Back to the Future franchise (Film), Battlestar Galactica (TV), Butterfly Effect (Film), The Caller (Film), Continuum (TV), Doctor Who (TV), Eureka (TV), The Final Count Down (Film), Frequency (Film), Fringe (TV), Paradox (TV), Sliders (TV, Sliders is not really about time travel but about travel between parallel universes through wormholes), Star Trek franchise (TV, Film), Stargate Atlantis (TV), Stargate SG-1 (TV), Strange Days at Blake Holsey High (TV), Terra Nova (TV), Terminator franchise (Film, TV), Thrill Seekers (Film), Timeline (Film), The Triangle (TV Miniseries), X-Files (TV), etc. The most intriguing idea in time travel stories is that one travels back in time and alter the future. Is that really possible?

What Do Physicists Say About Time Travel?

There isn’t really a physical theory of time travel, so there is no consensus among physicists as to whether time travel is possible. But they mostly agree that traveling back in time is impossible. There is no physical or technical reason for this belief but it is mainly due to a paradox called grandfather paradox. Let us say you went back in time and killed your future grandfather before he ever made your father. Then of course you would not exist in the future. However, you, who traveled back in time, do exist therefore a paradox. In some TV shows or films, when something like this happens you would just disappear by some mysterious force of nature to self-correct herself from an anomaly. Note that grandfather paradox assumes that there is only one timeline which appears to be the case according to our perception. But who knows? J. B. S. Haldane (1892 – 1964) once said “The universe is not only stranger than we imagine, it is stranger than we can imagine.” Time may not be just one dimensional but may be multi-dimensional and if so, there may not be grandfather paradox after all.

On the other hand, Einstein’s theory of relativity allows us to travel forward in time. But it is a boring time travel.

Three Relativistic Effects

Albert Einstein (1879 – 1955)

Imagine that you are traveling in a spaceship. According to the theory of relativity, you will experience three effects as the spaceship accelerates.

• Time Delay

$$\Delta t’=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$

This means that your clock runs slower as the spaceship accelerates.

• Contraction of Length

$$\ell=\ell_0\sqrt{1-\frac{v^2}{c^2}}$$

This means the spaceship gets shorter in length as it accelerates.

• Increase of Mass

$$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$

This means that the spaceship gets heavier as it accelerates.

The time delay effect allows us to travel forward in time. Let us say you travel to a distant star which is 200 light years away from Earth at the speed of light. When you come back to Earth, you will realize that it is not Earth as you know. It will be future Earth 400 years later. No one that you once knew including all your loved ones will be alive to greet you. But for you only a few hours passed by due to acceleration (departure) and deceleration (turning and arrival). This is a very boring time travel if one sees it as a travel forward in time. In effect of time travel, it is not so different from cryogenic stasis though you are not in sleep while traveling. By the way, if you can travel at the speed of light, you won’t age while you are traveling, so you can literally live forever. Unfortunately, this won’t happen to anyone no matter how advanced our science and technology would become. As shown earlier, as your spaceship accelerates it gets heavier and heavier making it harder to accelerate more. This is not something we can overcome by advancement of technology, but is simply part of nature. The only things that can travel at the speed of light are particles whose rest mass is 0 (massless particles) such as photons.

A Rebellion: Gödel Universe

Kurt Gödel (1906 – 1978)

Inspired by Einstein’s theory of general relativity, Kurt Gödel considered a model of universe that contains closed timelike curves. So, his model allows travel backward in time. Gödel also believed that one may travel back in time, influence past events and alter the future as a consequence. While Gödel universe is a solution of Einstein’s gravitational field equation, most physicists consider the model as unphysical.

Time Machine

It appears that at least at conceptual level the only thing that prevents us from traveling back in time is grandfather paradox or its variants. For now, let us forget about grandfather paradox. Is it then even remotely possible to build a time machine? The answer is affirmative and our time machine is called a wormhole. So what is a wormhole? A wormhole is a tubelike region in space-time which could be used as a shortcut to travel to a distant galaxy (as we have seen in the TV show Stargate) or could be used as a time machine. Wormholes are also solutions of Einstein’s gravitational field equation, but at the moment they are only mathematical solutions and there is no known physical reason as to why they must exist, nor is there known mechanism as to how they can be created. Although it is only a speculation, a wormhole may be created from a massive black hole.

Black Holes, White Holes, and Wormholes

When a massive star dies, it collapses due to the influence of its own gravity. This phenomenon is known as gravitational collapse. If the star is very massive, gravitational collapse leads to a singularity in space-time called, a black hole. It is called a black hole because even light is sucked into the black hole making it completely invisible to our naked eyes. According to Einstein’s general theory of relativity, gravitational field bends space-time around it i.e. it not only bends space but also bends time. As a result, there is time delay effect due to the presence of gravitational field: the stronger the gravitational field is the slower your clock runs. When a black hole is created, due to its massive gravitational field, your clock runs slower as you get closer to the black hole and it will eventually stop as you approach its event horizon. (Event horizon is point of no return. Once you pass through even horizon, there is no turning back, no way out of the black hole.) Since the space-time has no edges, the trajectories of particles fell into the event horizon of a black hole must be continued unless particles hit the black hole singularity. This requirement hints us that there may be an exit for these particles, the black hole’s counterpart, called a white hole and that the black hole and its counterpart white hole are connected by a tubelike region called a Lorentzian wormhole (also called a Schwarzchild wormhole or Einstein-Rosen bridge). We do not know yet what happens inside a black hole once objects enter its event horizon. So this is only a speculation but past the singularity time can be bent in either future or past direction. In other words, the white hole may appear somewhere at some time i.e. at an event in the future or at an event in the past. I will explain later but due to causality I claim that time can be bent only in past direction. I also speculate that warping space and time would result fluctuations in the space-time and when two local regions of the space-time get close enough, there can be an attractive force (a reminiscence of Casimir effect) between the two regions causing the creation of a wormhole connecting to a white hole at an event in the past.

A Lorentzian Wormhole

Causality

Every effect is followed by a cause. This is called causality in physics. For us, the future does not exist yet and so due to causality a wormhole cannot connect to an event in the future. Hence, we cannot travel forward in time through a wormhole. But the creation of a white hole at an event in the past also violates causality since the white hole did not exist at that event in the past. How do we cope with the change of history?

Chronology Protection Conjecture

Regarding time travel there are currently four different conjectures.

• The radical rewrite conjecture: One can travel forward and back in time and rewrite the history. (All Hell breaks loose case.)
• Novikov’s consistency conjecture: The Universe is consistent, so whatever temporal transpositions and trips one undertakes, events must conspire in such a way that the overall result is consistent.
• Hawking’s chronology protection conjecture: The cosmos works in such a way that time travel is completely and utterly forbidden. This conjecture permits spacewarps and wormholes but forbids timewarps and time machines.
• The boring physics conjecture: There are no wormholes nor spacewarps. There are no timewarps nor time machines.

Protect History at All Cost!

I am personally fond of Novikov’s consistency conjecture. In fact, I require a stronger version of Novikov’s consistency conjecture which does permit timewarps and time machines but does not permit any violation of causality nor any tampering with already recorded history. I speculate that as soon as any tampering with recorded history occurs such as the appearance of a white hole at some time and place in the past caused by a timewarp from the future, a new timeline must be created in order to prevent history from being rewritten. As a consequence, a new reality is created from that moment completely separately from the previously existing reality. This new reality may turn out to be pretty much similar to the previous reality, or it may turn out to be very different. Since timeline is split, whatever you do once you travel back in time through the wormhole, your action would only influence this new reality but not the original reality you are from. There will be no grandfather paradox or its variants and therefore there is nothing that will prevent you from traveling back in time and messing around with the past (new reality) such as killing your future grandfather or even killing your other yourself in this new reality. (Why anyone with right state of mind would do that though?) Suppose that I am right that the creation of a black hole leads to a timewarp into the past. Ever since the universe was created 15 billion years ago, many black holes must have been created. With my consistency conjecture, I can assert that there have already been many parallel universes (different realities) resulted from timewarps and that the universe we are living in (our reality) is very much likely not the one that was created for the first time. In other words, we are living in just one of many alternate realities.

Parallel Universes and Many-Time Physics

The creation of new realities results an emergence of new physics which I call many-time physics. If there are many different realities (parallel universes), the ambient space for the theory of physics should be $\mathbb{R}^{3+q}$ where $q>1$ is the time dimension. This new theory must be consistent with the current theory of physics on each reality i.e. the 4-dimensional space-time $\mathbb{R}^{3+1}$. It would be really interesting to see if different realities can interact with each other.

Tachyons

Tachyons are hypothetical FTL (Faster Than Light) particles whose notion appears a lot in science fiction. But the notion of FTL particles did not really come from science fiction. Tachyons were originally introduced by a Columbia University physicist Gerald Feinberg in 1967 as particles with imaginary rest mass. He thought they were FTL particles but it turned out that such particles do not propagate faster than light. Tachyons still appear in physical theories such as string theory. But they meant to be particles with imaginary rest mass. Here I use tachyons as generic name for FTL particles as in science fiction. These tachyons, if they exist, would not violate the theory of relativity as their initial velocity has already exceeded the speed of light when they were born. But there is no physical reason as to why they must exist or how they can be created. Tachyons are, if they exist, believed to travel back in time. Tachyons could be produced as a byproduct when a new reality is created because of a violation of causality. In fact, they may be the ordinary particles that have survived the singularity and traveled instantly from the future of one reality to the past of another reality through the wormhole. If so, it may be possible that one sends information from the future of one reality to the past of another reality through the wormhole, a message from the future!

Déjà Vu and Premonition: Messages from an Alternate Reality?

When you walk in the street you saw someone and you can be certain that you have never met the person in your life before. But the person looks so familiar and you cannot shake off the feeling that you somehow know the person. Or, you went to some place you have never been to before but everything looks so familiar and you cannot shake off the feeling that you know the place against the fact. What if it were true? It is just that it didn’t happen in this reality but it did happen in an alternate reality. Déjà vu or premonition that you are experiencing may be the messages you are receiving from the future of a different reality. Even if it were true and you were having a premonition that something bad might happen, you may not have to be serious about that though. What may have already happened in the future of a different reality may not necessarily happen in this reality. The reason for this is that every effect is followed by a cause (causality). Some causes are influenced by the decisions we make and some are random causes. Under the same circumstances, we may likely make the same decision (though I don’t believe it will always be the case), however there is no guarantee that there will be the same random causes at the same time and place in different realities. Otherwise, they wouldn’t be random. Someone like Michel de Nostredame (in French) or Nostradamus (in Latin) as more commonly known might have been tuned to receive messages from the future of another reality. Some events he described in his cryptic quatrains did actually occur but some didn’t including the grand quatrain about the end of the world that supposedly would (or could) have happened in 1999. For whatever reason, it didn’t happen, though it may have actually happened in another reality. How about the Mayan Calendar? Were they wrong? It may be possible that the Mayans were told by some beings (for them Gods) that the end of the world will come in the year 2012 and they believed it (of course because Gods said so). What if those beings the Mayans believed as Gods were time travelers from a distant future? They knew the world will end in 2012 and gave a warning to the Mayans because that is what actually happened in their reality. For some reason we don’t know (and we will perhaps never know), it did not happen in this reality.

J-Rod, a Grey Alien

There is an intriguing story floating around on the Internet about a grey alien called J-Rod. Before I continue, I want to make clear that I have absolutely no intention to get into Alien/UFO conspiracy theories. The reason I am trying to bring up a story about J-Rod is that there are some interesting aspects about this story of J-Rod in connection to my previous discussions. The story about J-Rod goes like this. 6 years after the famous Roswell crashes, there was another UFO crash near Kingman, Arizona in 1953. There were four grey aliens who survived the crash. One of them is J-Rod. This J-Rod worked as a technical adviser to the ultra-secret programs at Area 51. J-Rod does not speak and he communicated with his human collaborators only by telepathy. It turns out that he is not an alien but a human time traveler (or what has become of a human-being) from 50,000 years into the future. The whole race is facing dire medical conditions due to genetic mutations that threaten the entire civilization. (It sounds pretty much similar to the story about Asgards in Stargate or Stargate picked up the story of J-Rod?) They probably tried everything they could to save their dying race and none worked. The final solution they came up with is to come to our present time and harvest genetic materials from us (and even from other animals) to help repair damages in their bodies. (Perhaps this is the reason the greys have been abducting people and doing experiments on them against their will.)

Portals for Time Travel in the Milky Way Galaxy

According to J-Rod, there are naturally formed wormholes lurking in the Milky Way galaxy. I don’t know if they traveled through a naturally formed wormhole or an artificial wormhole or if they actually have ability to create one. Can one actually create an artificial wormhole like the one in Stargate? I don’t know the answer, but I can at least offer a speculation. Albert Einstein thought gravitation and electromagnetism are essentially the same force. Since he fled to America from Hitler’s Nazi regime in Germany, he had devoted the rest of his life to find the unified field theory that combines gravitation and electromagnetism. In spite of his efforts, he failed to find one, or did he? Modern physicists think that Einstein was wrong about his idea of the unified field theory. Regardless of a similarity between gravitation and electromagnetism as both being inverse-square laws, physicists don’t think they are the same force, at least not in this current universe (4 dimensional space-time). However, they believe that gravitation and electromagnetism along with the rest of known forces in the universe, weak and strong forces were the same one force before the Big Bang 15 billion years ago when the universe was 10 (or higher) dimensional space-time (Superstring Theory). But what if Einstein were right? There is a conspiracy theory that claims that Einstein indeed succeeded unifying gravitation and electromagnetism, and this led to the secret US military experiment called the Philadelphia Experiment carried out some time around October 28, 1943. The experiment was an attempt to cloak the U.S. Navy destroyer escort USS Eldridge by warping space using a strong magnetic field. Light would be bent around the ship along the warped space, so that the ship would become completely invisible to the enemy devices. But there was an unintended consequence. A wormhole was created and the ship was instantly teleported from the Naval Shipyard in Philadelphia to Norfolk, Virginia, over 200 miles away through the wormhole. If it were true, one can mimic gravitational field using strong magnetic field. Moreover, one may also create antigravity using electromagnetic field. While a naturally formed wormhole by a black hole is thought to be only one-way traffic, it may be possible to build an artificial wormhole that allows a time travel back and forth between the future and the past. Those folks who came from the future may be able to go back to the future through the artificial wormhole. How about us who are living in the present time? I am not so sure what might happen. There can be two different possible theories as to what might happen. One theory is that their future is not ours, so there will be no violation of causality and we also can travel to their future. Another theory is that although it is not our future, what happened in that reality might likely happen in our reality as well. So, if one travels to their future and comes back to our reality with knowledge from their future which may influence the course of our future, it could may well be a violation of causality. For that the Mother Nature may not allow us to travel to their future.

Time traveler with cell phone in 1928?

The Mayan Calendar and the End of the World in 2012?

According to J-Rod, something catastrophic happened in 2012 in their reality. It appears that there was some sort of solar superstorm at that time and their Earth got hit by the solar superstorm and their genetic mutations were caused by excessive amount of the radiation coming from the sun with the superstorm. Not all of them became like grey aliens.There is another group of humanoids, Nordic-looking humanoids. Both humanoid groups became technologically advanced but Nordic type humanoids became more spiritual. Earth eventually became inhabitable and both humanoid groups migrated to distant planets. Little grey type humanoids ended up on some planet in Zeta Reticuli which is about 39 light years away from Earth. Nordic type humanoids ended up on some planet in Orion’s Belt. This may indicate that those two groups of humanoids did not get along well with each other. It is possible that Mayans knew that the world (as we know it) will end in 2012 because they were told by these extraterrestrial beings who might have been perceived as Gods to the Mayans. In any event, as everybody knows such a catastrophe didn’t happen in 2012 in our reality. As I said earlier, things happened in one reality do not necessarily happen in another reality. What most of us didn’t know back then was that we could have had such a catastrophe but it was miraculously averted: we nearly missed a solar superstorm, one of the strongest ever recorded for the past 150 years, in 2012. When a powerful coronal mass ejection (CME) tore through Earth orbit in July 2012, Earth wasn’t there. However, if the eruption happened a week ago, Earth would’ve been directly hit. It might have been a sheer luck but we were spared. Now the sun became eerily quiet though supposedly it should be at its peak activity and intriguingly the famous sun spots have mostly disappeared. Now that they know the same disastrous catastrophe didn’t happen in this reality and Earth is still nicely habitable planet, the greys might consider mass migration into this reality if they prefer Earth over the planet in Zeta Reticuli. If so, all the nations on Earth might have to give the greys humanitarian (?) amnesty. If not, oh well, they might just take over Earth by force and use us as specimen for their cure. Let us hope that is not going to happen.

For those who would like to learn more about Black Holes, Wormholes and Time Travel, I recommend the following two books.

1. Black Holes and Time Warps: Einstein’s Outrageous Legacy, Kip S. Thorne and Stephen Hawking (Foreword), W. W. Norton & Company, 1995 (for lay people)
2. Lorentzian Wormholes: From Einstein to Hawking, Matt Visser, AIP Series in Computational and Applied Mathematical Physics, 2008 (for people with a substantial background in mathematics and physics)

Posted in Time Travel | 4 Comments

## Real Structure, Fundamental Symmetry and Quantum Mechanics

In the standard Hermitian quantum mechanics, the space of states of a quantum mechanical system forms a complex Hilbert space in general. But often the space of states becomes a real vector space. When the space of states is a real vector space, an alternative quantum mechanics may be considered.

Let $\mathbb{V}$ be the complex vector space of states of a quantum mechanical system. Define a map $J: \mathbb{V}\longrightarrow \mathbb{V}$ by
$$J\psi=\bar\psi$$
for every $\psi\in \mathbb{V}$. Then $J$ is an anti-linear involution called a real structure on $\mathbb{V}$. A real structure on a complex vector space is a way to decompose the complex vector space $\mathbb{V}$ into the direct sum of two real vector spaces. Any $\psi\in \mathbb{V}$ may be written as
$$\psi=\psi^++\psi^-,$$
\begin{align*}
\psi^+&:=\frac{1}{2}(\psi+J\psi)=\frac{1}{2}(\psi+\bar\psi)=\psi_{\mathrm{re}},\\
\psi^-&:=\frac{1}{2}(\psi-J\psi)=\frac{1}{2}(\psi-\bar\psi)=i\psi_{\mathrm{im}}.
\end{align*}
$J$ satisfies the properties:
\begin{align*}
J\psi^+&=\psi^+,\\
J\psi^-&=-\psi^-.
\end{align*}
Hence, one obtains a direct sum of vector spaces
$$\mathbb{V}=\mathbb{V}^+\oplus \mathbb{V}^-,$$
where
\begin{align*}
\mathbb{V}^+&=\{\psi\in \mathbb{V}: J\psi=\psi\},\\
\mathbb{V}^-&=\{\psi\in \mathbb{V}: J\psi=-\psi\}.
\end{align*}
Both $\mathbb{V}^+$ and $\mathbb{V}^-$ are real vector spaces. $\mathbb{V}^+$ is isomorphic to $\mathbb{V}^-$ via the isomorphism $\psi\longmapsto i\psi$. So,
$$\dim_{\mathbb{R}}\mathbb{V}^+=\dim_{\mathbb{R}}\mathbb{V}^-=\dim_{\mathbb{C}}\mathbb{V}$$
if $\dim_{\mathbb{C}}\mathbb{V}<\infty$. Let $\mathbb{V}^+=\mathbb{V}_{\mathbb{R}}$. Then $\mathbb{V}^-$ can be written as $i\mathbb{V}_{\mathbb{R}}$, so $\mathbb{V}$ may be viewed as the complexification of $\mathbb{V}_{\mathbb{R}}$
$$\mathbb{V}_{\mathbb{R}}^\mathbb{C}=\mathbb{V}_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}.$$
The real structure $J$ may be used to define an inner product $\langle\ ,\ \rangle$ on $\mathbb{V}$: For any $\varphi,\psi\in \mathbb{V}$,
\begin{align*}
\langle\varphi,\psi\rangle&=\int (J\varphi)\psi dV\\
&=\int\bar\varphi\psi dV.
\end{align*}
This inner product is of course the standard Hermitian product known as Dirac braket $\langle\ |\ \rangle$ to physicists. Therefore, the real structure gives rise to the Hilbert space structure on $\mathbb{V}$ as in the standard Hermitian quantum mechanics.

In studying a quantum mechanical system, it is not necessary to use complex-valued state functions but one may instead use real-valued state functions. For example, let us consider the Hamiltonian for quantum harmonic oscillator
$$\hat H=-\frac{\hbar^2}{2m}\frac{\partial}{\partial x^2}+\frac{m}{2}\omega^2\hat x^2.$$
The stationary Schrödinger equation $\hat H\psi=E\psi$ determines the eigenstates
$$\psi_n(\xi)=2^{-\frac{n}{2}}\pi^{-\frac{1}{4}}(n!)^{-\frac{1}{2}}e^{-\frac{\xi^2}{2}}H_n(\xi),$$
where $\xi=\sqrt{\frac{m\omega}{\hbar}}x$, and eigenvalues (energies)
$$E_n=\left(n+\frac{1}{2}\right)\hbar\omega$$
for $n=0,1,2,\cdots$. $H_n(x)$ are the Hermite polynomials and they satisfy the parity relation
$$H_n(x)=(-1)^nH_n(-x).$$
The eigenstates $\psi_n(\xi)$ forms a real Hilbert space with the usual Hermitian product (Dirac braket).

This time, let $\mathbb{V}$ be the real vector space of states of a quantum mechanical system. Let $\varphi:\mathbb{R}^3\longrightarrow\mathbb{R}$. $\varphi$ is said to be even under parity if $\varphi(-\mathbf{r})=\varphi(\mathbf{r})$. $\varphi$ is said to be odd under parity if $\varphi(-\mathbf{r})=-\varphi(\mathbf{r})$. Any real-valued function defined on $\mathbb{R}^3$ may be written as the sum of an even function and an odd function. Any $\varphi(\mathbf{r})\in \mathbb{V}$ can be written as
$$\varphi(\mathbf{r})=\frac{1}{2}[\varphi(\mathbf{r})+\varphi(-\mathbf{r})]+\frac{1}{2}[\varphi(\mathbf{r})-\varphi(-\mathbf{r})].$$
$\frac{1}{2}[\varphi(\mathbf{r})+\varphi(-\mathbf{r})]$ is an even function and $\frac{1}{2}[\varphi(\mathbf{r})-\varphi(-\mathbf{r})]$ is an odd function. Define a map $J: \mathbb{V}\longrightarrow \mathbb{V}$ by
$$J\varphi(\mathbf{r})=\varphi(-\mathbf{r})$$
for every $\varphi(\mathbf{r})\in \mathbb{V}$ i.e. $J$ is the parity. Then $J$ is a linear involution. Any $\varphi(\mathbf{r})\in \mathbb{V}$ may be written as
$$\varphi(\mathbf{r})=\varphi^+(\mathbf{r})+\varphi^-(\mathbf{r}),$$
where
\begin{align*}
\varphi^+(\mathbf{r})&=\frac{1}{2}[\varphi(\mathbf{r})+J\varphi(\mathbf{r})]=\frac{1}{2}[\varphi(\mathbf{r})+\varphi(-\mathbf{r})],\\
\varphi^-(\mathbf{r})&=\frac{1}{2}[\varphi(\mathbf{r})-J\varphi(\mathbf{r})]=\frac{1}{2}[\varphi(\mathbf{r})-\varphi(-\mathbf{r})].
\end{align*}
We have
\begin{align*}
J\varphi^+(\mathbf{r})&=\varphi^+(\mathbf{r}),\\
J\varphi^-(\mathbf{r})&=-\varphi^-(\mathbf{r}).
\end{align*}
The map $J$ is called the fundamental symmetry (see [1] for example). One gets a direct sum of vector spaces
$$\mathbb{V}=\mathbb{V}^+\oplus \mathbb{V}^-,$$
where
\begin{align*}
\mathbb{V}^+&=\{\varphi\in \mathbb{V}: J\varphi=\varphi\},\\
\mathbb{V}^-&=\{\varphi\in \mathbb{V}: J\varphi=-\varphi\}.
\end{align*}
If $\varphi(\mathbf{r})$ is even, then $\nabla\cdot\varphi(\mathbf{r})$ is odd and if $\varphi(\mathbf{r})$ is odd, then $\nabla\cdot\varphi(\mathbf{r})$ is even. If we say two functions $\varphi_1(\mathbf{r})$ and $\varphi_2(\mathbf{r})$ are identical if $\varphi_1-\varphi_2$ is constant, then $\mathbb{V}^+$ is isomorphic to $\mathbb{V}^-$ via the linear map $\nabla\cdot\varphi(\mathbf{r})$.

The fundamental symmetry may be used to define an inner product $\langle\ ,\ \rangle$ on $\mathbb{V}$ as follows: For any $\varphi(\mathbf{r}),\psi(\mathbf{r})\in \mathbb{V}$,
\begin{align*}
\langle\varphi(\mathbf{r}),\psi(\mathbf{r})\rangle&=\int (J\varphi(\mathbf{r}))\psi(\mathbf{r})dV\\
&=\int\varphi(-\mathbf{r})\psi(\mathbf{r})dV.
\end{align*}
For any $\varphi^+(\mathbf{r})\in \mathbb{V}^+$, $\varphi^-(\mathbf{r})\in \mathbb{V}^-$, we have
\begin{align*}
\langle\varphi^+(\mathbf{r}),\varphi^+(\mathbf{r}\rangle&=\int[\varphi^+(\mathbf{r})]^2dV\geq 0,\\
\langle\varphi^-(\mathbf{r}),\varphi^-(\mathbf{r})\rangle&=-\int[\varphi^-(\mathbf{r})]^2dV\leq 0
\end{align*}
We may assume that each $\varphi(\mathbf{r})$ is separable i.e. $\varphi(\mathbf{r})=X(x)Y(y)Z(z)$. ($\varphi(\mathbf{r})$ is a solution of the stationary Schrödinger equation $\hat H\varphi=E\varphi$ and this is a condition that can be imposed when we solve the equation.) Furthermore, we may assume that each of $X(x)$, $Y(y)$ and $Z(z)$ is either even or odd. Since $\varphi(-\mathbf{r})=X(-x)Y(-y)Z(-z)$, if $\varphi(-\mathbf{r})$ is odd, then at least one of $X(x)$, $Y(y)$, $Z(z)$ must be odd. For any $\varphi^+(\mathbf{r})\in \mathbb{V}^+$, $\varphi^-(\mathbf{r})\in \mathbb{V}^-$, $\varphi^+(\mathbf{r})\varphi^-(\mathbf{r})$ is odd. Let us write $\varphi^+(\mathbf{r})\varphi^-(\mathbf{r})=X(x)Y(y)Z(z)$ and say $X(x)$ is odd. Then
\begin{align*}
\langle\varphi^+(\mathbf{r}),\varphi^-(\mathbf{r}\rangle&=\int \varphi^+(\mathbf{r})\varphi^-(\mathbf{r})dV\\
&=\int_{-\infty}^\infty X(x)dx\int_{-\infty}^\infty Y(y)dy\int_{-\infty}^\infty Z(z)dz\\
&=0.
\end{align*}
Hence, $\varphi^+(\mathbf{r})$ and $\varphi^-(\mathbf{r})$ are orthogonal for all $\varphi^+(\mathbf{r})\in \mathbb{V}^+$, $\varphi^-(\mathbf{r})\in \mathbb{V}^-$, and so the direct sum $\mathbb{V}^+\oplus \mathbb{V}^-$ can be replaced by the orthogonal sum $\mathbb{V}^+\dotplus \mathbb{V}^-$. Note that $\langle\ ,\ \rangle$ is indefinte, so the fundamental symmetry gives rise to a Krein space structure rather than a Hilbert space structure on $\mathbb{V}$. Here, we are using a particular fundamental symmetry which is the parity, however our discussion may be generalised in terms of an arbitrary fundamental symmetry $J$. Although it may seem redundant, physically the inner product needs to be redefined as
\begin{align*}
\langle\varphi(\mathbf{r}),\psi(\mathbf{r})\rangle&=\int\overline{J\varphi(\mathbf{r})}\psi(\mathbf{r})dV\\
&=\int\bar\varphi(\mathbf{r})\psi(\mathbf{r})dV
\end{align*}
for any $\varphi(\mathbf{r}),\psi(\mathbf{r})\in \mathbb{V}$. The reason is that in quantum mechanics $\langle\ ,\ \rangle$ often interacts with operators that may be complex.

In the above example of quantum harmonic oscillator, let $\psi_n^+(\xi)=\psi_{2n}(\xi)$ and $\psi^-_{2n+1}(\xi)$ for $n=0,1,2,\cdots$. Then we have $\langle\psi^+_m,\psi^+_n\rangle=\delta_{mn}$, $\langle\psi^-_m,\psi^-_n\rangle=-\delta_{mn}$, and $\langle\psi^+_m,\psi^-_n\rangle=0$. Let $\mathbb{V}^+$ and $\mathbb{V}^-$ be spaces spanned by the orthogonal bases $\mathcal{B}^+=\{\psi^+_n(\xi): n=0,1,2,\cdots\}$ and $\mathcal{B}^-=\{\psi^-_n(\xi): n=0,1,2,\cdots\}$, respectively. Then $\mathbb{V}=\mathbb{V}^+\dotplus \mathbb{V}^-$ is a Krein space.

When the state functions of a quantum mechanical system are real-valued, the state functions may form a real Hilbert space or a Krein space depending on the choice between the inner products we discussed above. Hence, one may expect to have a quantum theory alternative, in which case a Krein space arises, to the standard Hermitian quantum mechanics. This alternative quantum theory is PT-Symmetric Quantum Mechanics which was first introduced by Carl M. Bender. PT-symmetric quantum mechanics has been promoted as a new quantum theory which admits a certain type of complex non-Hermitian Hamiltonians, the so-called PT-symmetric Hamiltonians. I recently showed that PT-symmetric quantum mechanics is indeed a Hermitian quantum mechanics and that those complex non-Hermitian (they are actually Hermitian) Hamiltonians are not really physical. (They would result the violation of unitarity.) After all, once disregarding those unphysical complex Hamiltonians, this so-called PT-symmetric quantum mechanics is not that much different at all from the standard Hermitian quantum mechanics. I will write more about this some other time. In the meantime, interesting readers may be referred to my recent papers [2], [3] on the issue.

Update 1: To clarify, here I meant a PT-symmetric Hamiltonian being Hermitian by it being self-adjoint with respect to the indefinite Hermitian inner product above. That’s why I said “PT-symmetric quantum mechanics is a Hermitian quantum mechanics.”

Update 2: My stance on PT-symmetric quantum mechanics has changed as I have a much better understanding of what’s going on. I considered complex PT-symmetric Hamiltonians were unphysical due to physicists’ insistance on using definite inner product because they want to interpret $|\psi|^2$ as probability like the case in ordinary quantum mechanics. But if we can relax about the meaning of $|\psi|^2$ as not necessarily probability but something that needs to be preserved under time evolution, the issue of unitarity violation can be resolved and complex PT-symmetric Hamiltonians become physical. It is possible that the interpretation of $|\psi|^2$ being probability is valid only in the ordinary quantum mechanics and if there is a new formulation of quantum mechanics, perhaps it may require a new set of rules and interpretations.

References:

[1] János Bognár, Indefinite Inner Product Spaces, Springer-Verlag 1974
[2] Sungwook Lee, $PT$-symmetric quantum mechanics is a Hermitian quantum mechanics, arXiv:1312.7738 [quant-ph]
[3] Sungwook Lee, On Finite $J$-Hermitian Quantum Mechanics, arXiv:1401.5149 [quant-ph]