[talanum1 (OP)]

Wikipedia states that there are positive and negative frequency solutions to the Klein-Gordon Equation. See:
https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation#Action.

By E = hf this must lead to negative Energy? They state that one can prove that both negative and positive frequency solutions lead to positive Energy. Isn't this a contradiction?

[evan_au]

By E = hf this must lead to negative Energy?

Einstein was only calculating with positive frequencies.

If you include negative frequencies, maybe the full equation is E = h|f| ?
Where |f| represents the magnitude of the frequency?

They state that one can prove that both negative and positive frequency solutions lead to positive Energy.

Mathematically, negative-frequency solutions routinely occur for Fourier transform operations on wave functions with positive energy.

When you apply the inverse Fourier transform to the combination of positive and negative frequencies, you get back to a real-valued waveform with positive energy.

See example here: https://en.wikipedia.org/wiki/Fourier_transform#Example

[Colin2B]

In general the sign of a frequency can indicate the direction of wave travel. If you take the example of 2 cars travelling in opposite directions you wouldn’t say one had +ve energy and the other -ve energy, if they collide the 2 kinetic energies add not cancel out.

[talanum1 (OP)]

If you include negative frequencies, maybe the full equation is E = h|f| ?

I have an issue with choosing that modification arbitrarily when needed.

Maybe there is negative Energy and space cannot distinguish the two.

In general the sign of a frequency can indicate the direction of wave travel. If you take the example of 2 cars travelling in opposite directions you wouldn’t say one had +ve energy and the other -ve energy, if they collide the 2 kinetic energies add not cancel out.

We have E = hf ! Both cars have positive frequencies.

[Colin2B]

In general the sign of a frequency can indicate the direction of wave travel. If you take the example of 2 cars travelling in opposite directions you wouldn’t say one had +ve energy and the other -ve energy, if they collide the 2 kinetic energies add not cancel out.

We have E = hf ! Both cars have positive frequencies.

With a single car/wave the direction is always taken as positive and both Planck and Einstein were considering single direction.
With 2 cars/waves in opposite directions you get both + and -, but E is +ve hence as @evan says E = h|f|.
I must apologise however, because I notice you started asking about Klein-Gordon. In that case in QFT the -ve and +ve frequencies are related to the creation (-ve) and annihilation (+ve) operators and just show that the flow of energy in each case is in opposite directions.

[evan_au]

By E = hf this must lead to negative Energy?

This equation came out of Einstein's work on the photoelectric effect (for which he won the Nobel Prize in 1923).

In fact the equation is:
K_max = hf - φ
Where:
- K_max is the maximum kinetic energy of photoelectrons emitted from a surface
- h is Planck's constant
- f is the frequency of the radiation
- φ is the work function of the material

So the domain of this equation is:
f > (K_max + φ)/h
Since the right side is positive, the left side (frequency) must also be positive.

Even with a hypothetical material where φ=0, the domain of this equation is f>0.
- You can't apply it to negative frequencies, since they won't liberate any electrons.
- If you consider negative frequencies to be photons moving away from the surface, they won't liberate any electrons either

See graph at: https://en.wikipedia.org/wiki/Photoelectric_effect#Mathematical_description

[talanum1 (OP)]

That "E = h|f|" is a modification to physics. If the absolute value does not apply, Physics is paradoxical.

The following URL talks about antimatter as having negative Energy:

Time stamp: 34:18. Talk by: Sean Carroll - a theoretical Physicist.

[Malamute Lover]

That "E = h|f|" is a modification to physics. If the absolute value does not apply, Physics is paradoxical.

The following URL talks about antimatter as having negative Energy:

Time stamp: 34:18. Talk by: Sean Carroll - a theoretical Physicist.

The video is unavailable. However I am familiar with what Sean Carroll says and antimatter having negative energy is not it. He speaks of virtual particle pairs, one normal matter and one antimatter having positive and negative energy. (Which is which does not matter.) This way the net balance is zero and these quantum fluctuations do not violate conservation laws. . However antimatter created in particle collider events definitely has positive energy. Otherwise the mass energy balance would be radically different from what is observed.

[uppfylltaoa]

When you apply the inverse Fourier transform to the combination of positive and negative frequencies, you get back to a real-valued waveform with positive energy.

When you apply the inverse Fourier transform to the combination of positive and negative frequencies, you get back to a real-valued waveform with positive energy.

[talanum1 (OP)]

When you apply the inverse Fourier transform to the combination of positive and negative frequencies, you get back to a real-valued waveform with positive energy.

That does not rule out E = hf.