# Have anybody considered travel with faster-than-infinite speeds?

Anixx
Logically, it seems that if we arrive at the point of destination before we left the point of departure, we have traveled with greater-than-infinite speed.
So, I wonder whether anyone introduced faster than infinite speeds in a physical or mathematical theory.
Classically, it seems, for travel from point ##(x_0, t_0)## to ##(x_1,t_1)## along the ##x## spatial axis and ##t## temporal axis, the mean speed would be
$$s=(x_1-x_0){\mathcal {H}}\int_{t_1-t_0}^\infty \frac1{t^2} dt$$
But if we let the integral to be divergent without undertaking Hadamard regularization, the speed will be
$$s=(x_1-x_0)\int_{t_1-t_0}^\infty \frac1{t^2} dt=(x_1-x_0)\left((1-\operatorname{sgn}(t_1-t_0))\pi\delta(0)+ \frac{1}{t_1-t_0} \right)$$
Here ##\pi\delta(0)## formally represents the divergent integral ##\frac12 \int_{-\infty}^\infty \frac1{t^2} dt## in non-Hadamard-finite-part sense. In other words, intead of taking the Hadamard finite part integral and getting negative speed for negative travel time, we leave the integral divergent and get faster-than-infinite speed.
For instance, when a particle propagates in a direction, a virtual particle-antiparticle pair may arise ahead of it, with antiparticle moving in the direction to meet the original particle. Then the antiparticle may annihilate with the original particle, while the virtual particle becoming rea and travels in the original direction. This is how quantum tunneling works. For a side observer this may look like the distance between the pair birth and the annihilation points has been traveled by the particle faster than with infinite speed.

weirdoguy, davenn and PeroK

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In the classical context, where particles cannot disappear and reappear, we can have no concept of a particle arriving earlier than it departed. If particle P is at position x1 at time t1 and position x2 at time t2 < t1 then we say the particle has traveled from x2 to x1 in the period [t2,t1]. No matter which way around we arrange the times and positions, under the classical paradigm we interpret that as traveling from its location at the earliest time to its location at the latest time, which must involve finite speed.

In the quantum context, we lose distinguishability of identical particles, so we cannot identify a post-annihilation particle with a pre-annihilation one.

Anixx
Yes, you are right, in classical physics there is no faster-than-infinite speed. It quantum mechanics, it seems, it exists. But my question was more related to the philisophical concept of speed (we can model a world with faster than infinite travel in a computer game, for instance).

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But my question was more related to the philisophical concept of speed
I'm afraid we do not discuss any philosophy on PF. It leads nowhere, everybody has an opinion, few have knowledge, and nobody has any idea how an acceptable answer would look like.

Mentor
Infinite speed is a concept that was thrown out when Classical Mechanics was superseded by Special Relativity theory where no physical matter can go faster than the speed of light.

Some theorists have posited the notion of tachyons, particles that travel faster than light but never travel slower than light. However, they have never been observed to exist.

https://en.wikipedia.org/wiki/Tachyon

John Wheeler and later Feynman theorized that electrons can travel back in time and we would observe them as positrons.

https://en.wikipedia.org/wiki/One-electron_universe

The one-electron universe postulate, proposed by John Wheeler in a telephone call to Richard Feynman in the spring of 1940, is the hypothesis that all electrons and positrons are actually manifestations of a single entity moving backward and forward in time.
...

Feynman was struck by Wheeler's insight that antiparticles could be represented by reversed world lines, and credits this to Wheeler, saying in his Nobel speech:
"I did not take the idea that all the electrons were the same one from [Wheeler] as seriously as I took the observation that positrons could simply be represented as electrons going from the future to the past in a back section of their world lines. That, I stole!"[1]
Feynman later proposed this interpretation of the positron as an electron moving backward in time in his 1949 paper "The Theory of Positrons" later applied it to all production and annihilation of particle-antiparticle pairs, stating that "the eventual creation and annihilation of pairs that may occur now and then, is no creation nor annihilation, but only a change of directions of moving particles, from past to future, or from future to past."
With that said, I think we've pretty much answered the OP's question that infinite speeds have been superseded by Special Relativity, and now it's time to close this thread.

Anixx