Professor Nambu had proposed a theory known as spontaneous symmetry breaking based on what was known to happen in superconductivity in condensed matter ; however, the theory predicted massless particles ( the Goldstone's theorem ), a clearly incorrect prediction.

Higgs " flash of inspiration " came while walking in the Cairngorms in 1964 and he wrote a short paper exploiting a loophole in Goldstone's theorem and published it in Physics Letters, a European physics journal edited at CERN, in Switzerland, later that year.

* Goldstone's theorem, examines a generic continuous symmetry

Goldstone's theorem examines a generic continuous symmetry which is spontaneously broken ; i. e., its currents are conserved, but the ground state ( vacuum ) is not invariant under the action of the corresponding charges.

A version of Goldstone's theorem also applies to nonrelativistic theories ( and also relativistic theories with spontaneously broken spacetime symmetries, such as Lorentz symmetry or conformal symmetry, rotational, or translational invariance ).

Hence the pions are massless,, in accordance with Goldstone's theorem.

The magnetization that occurs below is a famous example of the " spontaneous " breaking of a global symmetry, a phenomenon that is described by Goldstone's theorem.

For the second pilot, " Where No Man Has Gone Before " ( 1966 ), Roddenberry accepted director James Goldstone's decision to have Paul Fix play Dr. Mark Piper.

He had been brought in out of semi-retirement at Goldstone's recommendation at the last minute, after attempts to locate a cameraman had proved problematic.

Goldstone's work enabled multi-party negotiations to remain on course despite repeated outbreaks of violence, and his willingness to criticise all sides led to him being dubbed " perhaps the most trusted man, certainly the most trusted member of the white establishment " in South Africa.

Goldstone's work investigating violence led directly to him being nominated to serve as the first chief prosecutor of the United Nations International Criminal Tribunal for the former Yugoslavia and for Rwanda from August 1994 to September 1996.

The principle behind Goldstone's argument is that the ground state is not unique.

Prior to The Set-Up, Richard Goldstone's production credits had been limited to a half-dozen Our Gang comedy shorts.

* " Goldstone's Gaza Report, Part I: A Failure of Intelligence ", Middle East Review of International Affairs ( MERIA ), January 2010.

* " Goldstone's Gaza Report, Part II: A Miscarriage of Human Rights ," Middle East Review of International Affairs ( MERIA ), January 2010.

Practical applications are made impossible due to the no-cloning theorem, and the fact that quantum field theories preserve causality, so that quantum correlations cannot be used to transfer information.

Furthermore, Bob is only able to perform his measurement once: there is a fundamental property of quantum mechanics, known as the " no cloning theorem ", which makes it impossible for him to make a million copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results.

The spin-statistics theorem identifies the resulting quantum statistics that differentiates fermions from bosons.

The spin-statistics theorem holds that, in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions.

* Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily due to Wigner's theorem ( supersymmetry is another matter entirely ).

In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist.

The no-cloning theorem is a result of quantum mechanics that forbids the creation of identical copies of an arbitrary unknown quantum state.

Another way of stating the no-cloning theorem is that amplification of a quantum signal can only happen with respect to some orthogonal basis.

There is a classical analogue to the quantum no-cloning theorem, which we might state as follows: given only the result of one flip of a ( possibly biased ) coin, we cannot simulate a second, independent toss of the same coin.

The proof of this statement uses the linearity of classical probability, and has exactly the same structure as the proof of the quantum no-cloning theorem.

Thus if we wish to claim that no-cloning is a uniquely quantum result, some care is necessary in stating the theorem.

* The no-cloning theorem prevents us from using classical error correction techniques on quantum states.

In 1995, Shor and Steane revived the prospects of quantum computing by independently devising the first quantum error correcting codes, which circumvent the no-cloning theorem.

* The no-cloning theorem does not prevent superluminal communication via quantum entanglement, as cloning is a sufficient condition for such communication, but not a necessary one.

According to the spin-statistics theorem, particles with integer spin occupy symmetric quantum states, and particles with half-integer spin occupy antisymmetric states ; furthermore, only integer or half-integer values of spin are allowed by the principles of quantum mechanics.

Since, nonrelativistically, particles can have any statistics and any spin, there is no way to prove a spin-statistics theorem in nonrelativistic quantum mechanics.

Using Wick theorem on the terms of the Dyson series, all the terms of the S-matrix for quantum electrodynamics can be computed through the technique of Feynman diagrams.

In quantum field theory, the Weinberg – Witten theorem places some constraints on theories of composite gravity / emergent gravity.

The Reeh-Schlieder theorem of quantum field theory is sometimes seen as an analogue of quantum entanglement.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

* The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

It was introduced in 1971 by Stephen Cook in his seminal paper " The complexity of theorem proving procedures " and is considered by many to be the most important open problem in the field.

The corresponding form of the fundamental theorem of calculus is Stokes ' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily conserved and free from sources or sinks, having net flux equal to zero through any open domain.

It is used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences ( Chinese remainder theorem ) or multiplicative inverses of a finite field.

The theory of field extensions ( including Galois theory ) involves the roots of polynomials with coefficients in a field ; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel – Ruffini theorem on the algebraic insolubility of quintic equations.

Wedderburn's little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field.

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed ( with respect to the Krull topology below ) subgroups of the Galois group correspond to the intermediate fields of the field extension.

proved the theorem ( for the special case of polynomial rings over a field ) in the course of his proof of finite generation of rings of invariants.

Another version of Hahn – Banach theorem states that if V is a vector space over the scalar field K ( either the real numbers R or the complex numbers C ), if is a seminorm, and is a K-linear functional on a K-linear subspace U of V which is dominated by on U in absolute value,

This theorem states that a moving observer ( relative to the aether ) in his " fictitious " field makes the same observations as a resting observers in his " real " field.