In term of a multiplicative constant there are two definitions of phase factor: the global factor and the local factor. A global phase factor in the form of some function exp(����) in term of the overall multiplicative constant of a wavefunction ��(��) has no physical significance since ��(��) and exp(����)��(��) represent the same quantum state. This is true if and only if ��(��) is single valued, continuous, differentiable, and square integrable. These are usually well demonstrated in any textbook discussing the basic developments of quantum mechanics. On the other hand, in term of a local phase factor that multiplies given terms in an expansion coefficient of ��(��)=∑(��,��,��,��,��,��,��,…) ��(��,��,��,��,��,��,��,…) it does change physical prediction and represents changes in the quantum states. In other words, if ��(��) is expanded into ����(��)+����(��)+ ����(��)+ ����(��)+ ����(��)+ ����(��)+ ����(��)+… and then if ����(��) is replaced by a local gauge exp(����) ����(��) then clearly the square of the absolute value of ��(��) in the first expansion is not the same as that of the second expansion coefficient, although ����(��) and exp(����) ����(��) represent the same quantum state of local infinitesimal reality.

However, in a quantum theory of the spacetime continuum, these wavefunctions are replaced by Hadamard matrices where their matrix elements are usually allowed to take on real values of positive and negative unity or a mixture of both where they are equivalent to Euler’s identity for ��=�� and exp(����)=-1 or exp(-����)=-1 while matrices whose elements are all identically zero are used only to represent the zero mass and neutral charge states of physical reality. The products of multiplication of Hadamard matrices can be used to represent the various mass states while their sums of integration can be used to represent the various charges states of physical reality. The important advantages of using Hadamard matrices as phase factors of cold fusion are the facts that integrating the same order matrices represents the states for charge neutrality and multiplying the same order matrices represents states of zero mass of physical reality.