Your post does not address the point raised by us. How do you define "sound mathematics?" What we meant to say is that the "mathematics" of physicists is unmathematical. It is more of a manipulation of numbers to suit one's convenience. A proof of this is the plethora of theories and interpretations that go in the name of Quantum Theory. Most of what is called as "mathematics" in modern science fails the test of logical consistency that is a corner stone for judging the truth content of a mathematical statement. For example, you do not define infinity - it is not a big number. All mathematical operations involving infinity is void. Thus, renormalization is not mathematical.
The SchrÃ¶dinger equation was devised to find the probability of finding the particle in the narrow region between x and x+dx, which is denoted by P(x) dx. The function P(x) is the probability distribution function or probability density, which is found from the wave function ψ(x) in the equation P(x) = [ψ(x)]2. The wave function is determined by solving the SchrÃ¶dinger's differential equation: d2ψ/dx2 + 8π2m/h2 [E-V(x)]ψ = 0, where E is the total energy of the system and V(x) is the potential energy of the system. By using a suitable energy operator term, the equation is written as Hψ = Eψ. The equation is also written as iħ ∂/∂t|ψâ€º = H|ψâ€º, where the left hand side represents iħ times the rate of change with time of a state vector. The right hand side equates this with the effect of an operator, the Hamiltonian, which is the observable corresponding to the energy of the system under consideration. The symbol ψ indicates that it is a generalization of SchrÃ¶dinger's wave-function. The way the equation has been written, it appears to be an equation in one dimension, but in reality it is a second order equation signifying a two dimensional field, as the original equation and the energy operator contain a term x2. The method of the generalization of the said SchrÃ¶dinger equation to the three spatial dimensions does not stand mathematical scrutiny. A third order equation implies volume. Addition of three areas does not generate volume and neither x+y+z ≠ (x.y.z) nor x2+y2+z2 ≠ (x.y.z). Thus, there is no wonder that it has failed to explain spectra other than hydrogen. The so-called success in the case of helium and lithium spectra gives results widely divergent from observation.
The probability calculations needed to work out the chance of finding an electron (say) in a particular place at a particular time actually depend on calculating the square of the complex number corresponding to that particular state of the electron. But calculating the square of a complex variable does not simply mean multiplying it by itself since it is not the same as a real number. Instead, another variable, a mirror image version called the complex conjugate is considered, by changing the sign in front of the imaginary part (if it was + it becomes - and vice versa). The two complex numbers are then multiplied together to give the probability. This shows that, truly it is not squaring, but a mathematical manipulation as the negative sign implies physical non-existence of the second term like the physical non-existence of a mirror image. If A has 5 apples and he gives it to B, then only B has those five apples and A is said to have -5 apples to signify his ownership of the five apples physically with B. Similarly, the mirror image does not make two objects, but only one real object and the other physically non-existent image. This is not mathematics, as mathematics deals with numbers, which is a characteristic of physical objects. Similarly, mathematically all operations involving infinity are void. Hence renormalization is not mathematical. The brute force approach where several parameters are arbitrarily reduced to zero or unity is again not mathematical, as the special conditions that govern the equality sign for balancing cause and effect are ignored. The arbitrary changes change the characteristic of the system. If we treat the length of all fingers as unity, then we cannot hold an object properly. There are innumerable instances of un-mathematical manipulation in the name of mathematics.
The requirement that all fundamental theories be presented within a concise mathematical framework virtually prevented serious theoretician from ever considering a non-field theory because of its mathematical complexities. The "mathematics" involved in field theories to describe events are simple and concise when compared with the "mathematics" of the same event in non-field terminology. Non-field theories are denied serious consideration because they cannot be given a precise mathematical description. Even if someone was able to develop a precise set of non-field equations, they would likely be so complex, mystifying and un-mathematical that only few mathematicians would be able to understand them.
Kindly clarify the points raised above.