Boost along thw parrallel beams. As the boost inncreases the energy-momentum density drops to zero. As any motion of the beams toward each other has to exist in all such frames and a it approaches zero as boost increases, it has to be zero in all frames.
Apart from technical mathematical cases (tensors) you might just consider 'scalar' to be a real number as opposed to a set of numbers. E.,g., px is a 'scalar' but it makes no sense to ask how it transforms other then in the context of being one element of a 4-vector. Energy is just the same...
QM will of course modify GR when and if we ever figure out how to make them consistent with each other. String/Brane theory claims to do this, but so far not in a very useful way. It conceivable that GR+QM will explain why the cosmological constant is so small, yet not zero, or may be not!
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Sure I can have it both ways. It's just words. Complex observable is just a word for needing two numbers to describe the measurement. It get's elegant when the multiplication is defined to form a 'ring' . I can _call _that measuring a complex variable. In the other sense all the numbers that get...
An 'imaginary' frequency is just a plain old exponential or a 'sinh' or something. Nothing unreal about it. It's just words. A frequency could be called an 'imaginary' value of a a decay or growth rate. All the numbers whether in pairs or not are, of course, numbers.
A complex number is just two real numbers with a rule for 'multiplying' them. A simple rate is just one real number. If it varies in time or space it takes more than one real number to describe the dependence. The measurements of dependence are just as real as anything. There's in fact even then...
https://skepticalsciencereviews.wordpress.com/reviews/
See "Are imaginary numbers real" at above site.
They are! They are two numbers. We measure 'em all the time in interference experiments. They are just as real as a tensor describing Electromagnetic fields. Nothing 'imaginary' about 'em.
In a more fundamental you can think about this w/o explicit reference to commutation rules. A 'particle' is an quantized excitation of a field. It is more wave than particle. The particle like-behavior arises when the excitation is in the form of a wave-packet which is localized, but to get a...
Try something a little less mathy. Remember Darwin was not that good at math.
Try something a little less ambitious. In physics experimentalist always make a contribution while theorist get the glory, but many of them contribute more noise than signal. If you understand how to do an integral...
Interesting thread.
One thing there are no particles in QFT. The electrons are not a particles, but excitations of the electron-field. Particle-like appearance is when the excitations form a packet (many wave lengths) that is localized.
In this sense the wave picture is more correct than the...