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A nonlocal quantum clock records quantum circuit time in an abstract $T+1$ dimensional register. A binary encoding uses logarithmically many qubits, but transitions $|t⟩⟨t-1|$ need not have bounded qubit locality. Extra diagonal penalties may exclude unused binary labels. A unary quantum clock provides a different encoding for fixed-locality constructions.

A Feynman-Kitaev Hamiltonian penalizes incorrect input initialization, disagreement between successive quantum circuit steps and clock labels, and optionally a rejecting output. Its propagation quadratic form is a sum of $\frac{1}{2}\Vert {\eta }_t-U_t{\eta }_{t-1}{\Vert }^2$. Without output penalty, its zero-energy space consists of correctly initialized computational history states. A nonlocal quantum clock makes the propagation formula simple but does not supply fixed qubit locality by itself.

Renormalization can relate bare operators to a matrix of renormalized operators, ${\mathcal{O}}_{i,B}=Z_{ij}{\mathcal{O}}_{j,R}$. The corresponding anomalous dimension matrix acts on each insertion slot in the Callan-Symanzik equation.

A contact term is a distribution supported where insertion points coincide. Its Fourier transform is local in momentum, typically polynomial. Renormalizing an operator product can require additive contact counterterms even if each operator has zero anomalous dimension.

The complex modulus of $z=x+iy$ is $|z|=\sqrt{x^2+y^2}=\sqrt{z\overline{z}}$. It is the Euclidean norm of $(x,y)$ in the complex plane, and satisfies $|zw|=|z||w|$ and the triangle inequality.

A nonzero complex number has polar formwhere $r$ is its modulus and $\theta$ is an argument. The equality uses Euler's formula; sine and cosine recover the imaginary part and real part from polar coordinates. For $z=3+4i$, take $r=5$ and $\theta =\arctan (4/3)\approx 0.927295$ radians, so $\cos \theta =3/5$ and $\sin \theta =4/5$.

If $w=s{e}^{i\varphi }\ne 0$, thenThus multiplication multiplies moduli and adds arguments modulo $2\pi$; division divides moduli and subtracts arguments modulo $2\pi$. Since $1+i=\sqrt{2}{e}^{i\pi /4}$, multiplying $3+4i$ by $1+i$ gives $-1+7i$, of modulus $5\sqrt{2}$ and argument $\theta +\pi /4\approx 1.712693$ radians. One formula simultaneously describes its new size and direction.

Repeated multiplication now becomes especially simple: $(r{e}^{i\theta }{)}^n=r^n{e}^{in\theta }$ for a positive integer $n$. This explains the evenly spaced arguments of the roots of a complex number. For example, $(3+4i{)}^2=-7+24i$ has modulus $25$ and principal argument $2\theta \approx 1.854590$ radians. The Cartesian calculation and the polar form provide independent checks on both its location and its size. Choose polar form when the task concerns multiplication, division, or repeated turns; use real parts and imaginary parts for addition.

For a bounded operator $K$ between Hilbert spaces, compactness is equivalent to $f_n\rightharpoonup f$ implying $K{f}_n\to Kf$ in norm. The Uniform boundedness principle bounds a weakly convergent sequence. Relative compactness of its images and the unique possible weak limit then give norm convergence. Conversely, every bounded domain sequence has a weakly convergent subsequence, so the image of the unit ball is relatively compact.

A comoving volume has fixed boundaries in comoving coordinates. Its physical volume grows as the cube of the scale factor. In a homogeneous adiabatic gas, cosmological entropy conservation makes $sV$ constant.

A collision cross-section is the effective area that converts a target number density and incident particle speed into a collision rate. It can depend on particle species, relative speed, and long-range forces, and need not equal a purely geometric collision cross-section.

A collision is an interaction between approaching bodies or particles that changes their motion. Momentum conservation and conservation of energy constrain the outcomes; inelastic collisions can transfer mechanical energy into internal degrees of freedom.

The conjugate momentum to a periodic internal soliton angle generates shifts of that angle. Single-valued wavefunctions, or Bohr-Sommerfeld quantization with zero Maslov index, quantize this canonical momentum. A theta term can shift the relation between canonical momentum and mechanical charge without changing integer canonical labels.

The first $a$ sets of a fixed size in colexicographic order form a colexicographic initial segment. Its lower shadow is again a colexicographic initial segment at the next smaller size.

A degree-$q$ cohomology group is the quotient of degree-$q$ cocycles by coboundaries in a cochain complex. Topological cohomology groups arise from singular or cellular cochains. Their direct sum carries the cohomology ring structure when equipped with the cup product.

The Borel version of equivariant cohomology is the ordinary cohomology of the Borel construction. It retains information about both the space and its group action. For a point it is the cohomology of the classifying space; for a free action with the usual bundle hypotheses it is the cohomology of the orbit space.

For a based space, reduced cohomology removes the coefficient-ring summand coming from the basepoint. In nonnegative degrees it is the kernel of restriction $H^{\ast }(X;R)\to H^{\ast }(\{x_0\};R)$. It agrees with ordinary cohomology in positive degrees and admits the suspension isomorphism and exact sequences for based cofibrations.

A cohomology operation is a natural transformation between cohomology functors. Thus for every continuous map $f$, its pullback commutes with the operation: $\theta (f^{\ast }a)=f^{\ast }\theta (a)$. This makes operations additional invariants of homotopy equivalence, beyond the additive groups and their cup products.

For a topological space $X$ and abelian group $A$, singular cohomology is the cohomology of the cochain complex $C^n(X;A)=\mathrm{Hom}(C_n(X;\mathbb{Z}),A)$, where $C_n$ is the singular chain group. Its differential is dual to the singular boundary. This theory has relative cohomology, long exact sequences and the Excision theorem. On spaces that are not locally contractible it can differ from Čech cohomology.

The thermal trace closes bosonic coherent-state paths periodically and fermionic coherent-state paths antiperiodically: $\psi (\beta )=\zeta \psi (0)$ and $\bar{\psi }(\beta )=\zeta \bar{\psi }(0)$, with $\zeta =1$ for bosons and $-1$ for fermions. The minus sign follows from the fermionic coherent-state trace, not from choosing a different Hamiltonian. These boundary conditions determine the allowed Matsubara frequencies.

A quantum channel on the receiving system cannot increase coherent information: $I(A⟩B{)}_{\rho }\ge I(A⟩B^{\prime }{)}_{\sigma }$. Dilate the channel to a linear isometry of Hilbert spaces $B\to B^{\prime }E$. The difference is $I(A:E|B^{\prime })$ of the dilated state, nonnegative by Strong subadditivity of Von Neumann entropy.

Purify the input on $RA$ and dilate a channel by $A\to BE$. The joint state $RBE$ is pure, so complementary entropies obey $S(B)=S(RE)$ and $S(RB)=S(E)$. Therefore coherent information is $I_c=S(B)-S(RB)=S(R\mid E)=-S(R\mid B)$. The positive environment conditional entropy and negative receiving-system conditional entropy are two equivalent forms.