In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the infinite-dimensional case of a linear operator S mapping a function space V to itself. The corresponding quantity det(S) is called the functional determinant of S.
There are several formulas for the functional determinant. They are all based on the fact that the determinant of a finite matrix is equal to the product of the eigenvalues of the matrix. A mathematically rigorous definition is via the zeta function of the operator,
where tr stands for the functional trace: the determinant is then defined by
where the zeta function in the point s = 0 is defined by analytic continuation. Another possible generalization, often used by physicists when using the Feynman path integral formalism in quantum field theory (QFT), uses a functional integration:
This path integral is only well defined up to some divergent multiplicative constant. To give it a rigorous meaning it must be divided by another functional determinant, thus effectively cancelling the problematic 'constants'.
These are now, ostensibly, two different definitions for the functional determinant, one coming from quantum field theory and one coming from spectral theory. Each involves some kind of regularization: in the definition popular in physics, two determinants can only be compared with one another; in mathematics, the zeta function was used. Osgood, Phillips & Sarnak (1988) have shown that the results obtained by comparing two functional determinants in the QFT formalism agree with the results obtained by the zeta functional determinant.
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In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order...
In mathematics, the determinant is a scalar value that is a certain function of the entries of a square matrix. The determinant of a matrix A is commonly...
mathematical functions Clifford algebra Tensor field Spectral theory Functionaldeterminant Fulton, William; Harris, Joe (1991). Representation Theory: A First...
divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly...
Hesse and later named after him. Hesse originally used the term "functionaldeterminants". The Hessian is sometimes denoted by H or, ambiguously, by ∇2...
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) ∞ {\displaystyle (2\pi )^{\infty }} is infinite and also, the functionaldeterminant would also be infinite in general. This can be taken care of if...
anomaly in quantum field theory. It uses the correspondence between functionaldeterminants and the partition function, effectively making use of the Atiyah–Singer...
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wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic energy functional of such a system is known exactly. The exchange–correlation...
classifying patients in class II or III, because functional capacity is such a powerful determinant of outcome, it remains arguably the most important...
)}(\eta (A))\det(\mathrm {d} \eta /\mathrm {d} A)\mathrm {d} A} . The functionaldeterminant and associated mathematical subtleties drop out if the Langevin...
equations. The first systematic methods for solving linear systems used determinants and were first considered by Leibniz in 1693. In 1750, Gabriel Cramer...
of R containing x. Customarily X is called the determinant set and Y the dependent set. A functional dependency FD: X → Y is called trivial if Y is a...
(x)\delta \phi (y)}}[B]\right)^{-1/2}+\cdots } where "det" signifies a functionaldeterminant and C is a constant. The power of minus one half will naturally...
Polyakov formula expresses the conformal variation of the zeta functionaldeterminant of a Riemannian manifold. Proposed by Alexander Markovich Polyakov...
perturbation takes place. The ghost term in the Lagrangian represents the functionaldeterminant of the Jacobian of this embedding, and the properties of the ghost...
Quillen metric Quillen, Daniel (1985), "Determinants of Cauchy-Riemann operators over a Riemann surface", Functional Analysis and Its Applications, 19, Springer...
to evolutionary ideas of the Enlightenment. Lagrange considers a functionaldeterminant of order 3, a special case of a Jacobian. He also proves the expression...
; Schrader, R.; Seiler, R. (1978). "A conjecture on the spinor functionaldeterminant". Nuclear Physics B. 142 (4): 525–534. Bibcode:1978NuPhB.142..525H...
appears naturally in connection with extremal problems for the functionaldeterminant of the Laplacian (via the Polyakov formula; see Branson & Ørsted...
only if its determinant is nonzero. Therefore, an alternative definition of GL(n, F) is as the group of matrices with nonzero determinant. Over a commutative...
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