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The S-matrix is closely related to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; the elements (individual numerical entries) in the S-matrix are known as scattering amplitudes. Poles of the S-matrix in the complex-energy plane are identified with bound states, virtual states or resonances.
The S-parameter matrix for the 2-port network is probably the most commonly used and serves as the basic building block for generating the higher order matrices for larger networks. [18] In this case the relationship between the outgoing ('reflected'), incident waves and the S-parameter matrix is given by:
S-matrix theory was a proposal for replacing local quantum field theory as the basic principle of elementary particle physics. It avoided the notion of space and time by replacing it with abstract mathematical properties of the S -matrix .
In quantum field theory, correlation functions, often referred to as correlators or Green's functions, are vacuum expectation values of time-ordered products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as S-matrix elements.
The S-matrix can be written as: = + where is the part of the S-matrix that is due to interactions; e.g. = just implies the S-matrix is 1, no interaction occur and all states remain unchanged. Unitarity of the S-matrix: † = is then equivalent to:
A gate that acts on qubits (a register) is represented by a unitary matrix, and the set of all such gates with the group operation of matrix multiplication [a] is the unitary group U(2 n). [2] The quantum states that the gates act upon are unit vectors in 2 n {\displaystyle 2^{n}} complex dimensions, with the complex Euclidean norm (the 2-norm ).
where Z is an N × N matrix the elements of which can be indexed using conventional matrix notation. In general the elements of the Z-parameter matrix are complex numbers and functions of frequency. For a one-port network, the Z-matrix reduces to a single element, being the ordinary impedance measured between the two terminals. The Z-parameters ...
This is nowadays known not to be true, since there are many theories which are nonperturbatively consistent, each with their own S-matrix. Without the narrow-resonance approximation, the bootstrap program did not have a clear expansion parameter, and the consistency equations were often complicated and unwieldy, so that the method had limited ...