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These points may be joined forming a shape like a spider. Joined points represent an "or" condition, also known as a logical disjunction. A spider diagram is a boolean expression involving unitary spider diagrams and the logical symbols ,,. For example, it may consist of the conjunction of two spider diagrams, the disjunction of two spider ...
An example ZX-diagram. This one has two inputs (wires coming from the left), and three outputs (wires exiting to the right), and hence it represents a linear map from to . ZX-diagrams consist of green and red nodes called spiders, which are connected by wires. Wires may curve and cross, arbitrarily many wires may connect to the same spider, and ...
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U −1 equals its conjugate transpose U *, that is, if = =, where I is the identity matrix.. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † ), so the equation above is written
From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group 2 A n, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group (reversing the Dynkin diagram A n, which corresponds to transpose inverse) and the field automorphism of the extension ...
More formally, in the context of QFT, the S-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of physical states: a multi-particle state is said to be free (or non-interacting) if it transforms under Lorentz transformations as a tensor product ...
On the other hand, if is compact, then every finite-dimensional representation of admits an inner product with respect to which is unitary, showing that decomposes as a sum of irreducibles. [9] Similarly, if g {\displaystyle {\mathfrak {g}}} is a complex semisimple Lie algebra, every finite-dimensional representation of g {\displaystyle ...
the unitary group U(n) and the special unitary group SU(n), the compact forms of the exceptional Lie groups: G 2, F 4, E 6, E 7, and E 8. The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples (which already includes some redundancies). This classification is ...
The free field model can be solved exactly, and then the solutions to the full model can be expressed as perturbations of the free field solutions, for example using the Dyson series. It should be observed that the decomposition into free fields and interactions is in principle arbitrary.