Can infinity + infinity = infinity?
The answer is definite no if by the above mathematical operation one means simple linear summation. Simply speaking, the argument that 1 + 1 = 2 and 2 + 3 = 5 cannot be extended and extrapolated to infinity.
But if it means direct sum being done in Hilbert space in quantum mechanics and relativistic quantum field theories then it is possible. Hilbert space is a subspace of real physical space and it is in this subspace that realms of quantum mechanics take shape.
The fact is that the sum of two Dirac Delta function could be another Dirac Delta function, in simple R1-to-R4 spaces or in Hilbert space. Or two infinitely spooked functions at a particular space-time point or at different ones, and zero elsewhere could result in a finite function for all coordinates.
The same is the case with direct product in Hilbert space. I do not know anything as direct subtraction or direct division in Hilbert space to the best of my knowledge. May be with the invention of newer phenomenon they are required to be defined.
But as a matter for argument, Infinity is a concept and not a measure of anything. To be correct infinity is not a number and can never be put on a number scale. Any argument using putting infinity on number scale to prove the existence of God is totally incorrect or minimally incomplete non-cogent argument.
Formally, there is no measure greater than +infinity and no measure smaller than infinity, but still + and - infinities are no measures in real time. All physical quantities, including space-time, have to be finite. The simple addition operation is not a defined operation on two infinities and the same is true with subtraction, division, multiplication and other simple linear mathematical operations. However, some higher mathematical operations could be performed on infinities under certain circumstances.
The one easily known example occurs in quantum field calculations, whereby through a process called renormalisation a finite measure is obtained by subtracting two infinities, i.e.
one infinity second infinity = finite measurable number of physical quantities as in relativistic quantum electrodynamics calculations of physical variables, like gyromagnetic ratio of Leptons and Baryons; g/2 etc.
Infinity here comes through a physical concept known as the self energy. Self energy manifests itself in loops in what is known as Feynman diagrams and they can be in multiple numbers in some of the diagrams. Higher the number of loops, lower is the contribution of that term in calculations and it is equally difficult to get rid of various inherent infinities. But then it is not a formal subtraction process and involves complicated relativistically invariant Complex Green Functions and higher mathematical concepts.
Therefore, subtraction is an abuse of the word and normal, simple arithmetic operations cannot be defined on infinity. The same is case with addition. Again, infinity is not a measure and like summation, subtraction and other simple linear mathematical operations on infinities should be considered as wrong.
But yes two infinities could be added to give infinity or even a finite quantity. The examples may include calculations involving during the explosion of Big Bang, crunching of the Universe, expansion of Black Holes and White Holes, collision of two or more Universes etc. But then higher mathematics is required to work out such additions. May be in super-symmetric renormalized relativistic quantum field theories of gravitation, the additions of two infinities could give another infinity or even finite results.
Please spare me from negative comments, if anyone finds my arguments to be totally incorrect. As always, I do not claim that I understand science.
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