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118-452: In particle physics , CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry ): the combination of C-symmetry ( charge conjugation symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C-symmetry) while its spatial coordinates are inverted ("mirror" or P-symmetry). The discovery of CP violation in 1964 in

236-2117: A → 2 → b {\displaystyle a\rightarrow 2\rightarrow b} . This is exactly the case for the kaon where the decay is performed via different quark channels (see the Figure above). In this case we have: M = | M 1 |   e i θ 1   e i ϕ 1 + | M 2 |   e i θ 2   e i ϕ 2 M ¯ = | M 1 |   e i θ 1   e − i ϕ 1 + | M 2 |   e i θ 2   e − i ϕ 2   . {\displaystyle {\begin{alignedat}{3}{\cal {M}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{i\phi _{2}}\\{\bar {\cal {M}}}&=|{\cal {M}}_{1}|\ e^{i\theta _{1}}\ e^{-i\phi _{1}}&&+|{\cal {M}}_{2}|\ e^{i\theta _{2}}\ e^{-i\phi _{2}}\ .\end{alignedat}}} Some further calculation gives: | M | 2 − | M ¯ | 2 = − 4   | M 1 |   | M 2 |   sin ⁡ ( θ 1 − θ 2 )   sin ⁡ ( ϕ 1 − ϕ 2 ) . {\displaystyle |{\cal {M}}|^{2}-|{\bar {\cal {M}}}|^{2}=-4\ |{\cal {M}}_{1}|\ |{\cal {M}}_{2}|\ \sin(\theta _{1}-\theta _{2})\ \sin(\phi _{1}-\phi _{2}).} Thus, we see that

354-487: A Hilbert space , which is also treated in quantum field theory . Following the convention of particle physicists, the term elementary particles is applied to those particles that are, according to current understanding, presumed to be indivisible and not composed of other particles. Ordinary matter is made from first- generation quarks ( up , down ) and leptons ( electron , electron neutrino ). Collectively, quarks and leptons are called fermions , because they have

472-502: A quantum spin of half-integers (−1/2, 1/2, 3/2, etc.). This causes the fermions to obey the Pauli exclusion principle , where no two particles may occupy the same quantum state . Quarks have fractional elementary electric charge (−1/3 or 2/3) and leptons have whole-numbered electric charge (0 or 1). Quarks also have color charge , which is labeled arbitrarily with no correlation to actual light color as red, green and blue. Because

590-708: A set of measure zero . The inner product of functions f and g in L ( X , μ ) is then defined as ⟨ f , g ⟩ = ∫ X f ( t ) g ( t ) ¯ d μ ( t ) {\displaystyle \langle f,g\rangle =\int _{X}f(t){\overline {g(t)}}\,\mathrm {d} \mu (t)} or ⟨ f , g ⟩ = ∫ X f ( t ) ¯ g ( t ) d μ ( t ) , {\displaystyle \langle f,g\rangle =\int _{X}{\overline {f(t)}}g(t)\,\mathrm {d} \mu (t)\,,} where

708-1058: A " Theory of Everything ", or "TOE". There are also other areas of work in theoretical particle physics ranging from particle cosmology to loop quantum gravity . In principle, all physics (and practical applications developed therefrom) can be derived from the study of fundamental particles. In practice, even if "particle physics" is taken to mean only "high-energy atom smashers", many technologies have been developed during these pioneering investigations that later find wide uses in society. Particle accelerators are used to produce medical isotopes for research and treatment (for example, isotopes used in PET imaging ), or used directly in external beam radiotherapy . The development of superconductors has been pushed forward by their use in particle physics. The World Wide Web and touchscreen technology were initially developed at CERN . Additional applications are found in medicine, national security, industry, computing, science, and workforce development, illustrating

826-548: A Hilbert space that, with the inner product induced by restriction , is also complete (being a closed set in a complete metric space) and therefore a Hilbert space in its own right. The sequence space l consists of all infinite sequences z = ( z 1 , z 2 , ...) of complex numbers such that the following series converges : ∑ n = 1 ∞ | z n | 2 {\displaystyle \sum _{n=1}^{\infty }|z_{n}|^{2}} The inner product on l

944-554: A complex phase causes CP violation (CPV) is not immediately obvious, but can be seen as follows. Consider any given particles (or sets of particles)   a   {\displaystyle \ a\ } and   b   , {\displaystyle \ b\ ,} and their antiparticles   a ¯   {\displaystyle \ {\bar {a}}\ } and   b ¯   . {\displaystyle \ {\bar {b}}\ .} Now consider

1062-603: A complex phase gives rise to processes that proceed at different rates for particles and antiparticles, and CP is violated. From the theoretical end, the CKM matrix is defined as   V C K M = U u † U d {\displaystyle \ V_{\mathrm {CKM} }=U_{u}^{\dagger }U_{d}} , where U u {\displaystyle U_{u}} and U d {\displaystyle U_{d}} are unitary transformation matrices which diagonalize

1180-592: A definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory. Lebesgue spaces are function spaces associated to measure spaces ( X , M , μ ) , where X

1298-507: A distance function defined in this way, any inner product space is a metric space , and sometimes is known as a pre-Hilbert space . Any pre-Hilbert space that is additionally also a complete space is a Hilbert space. The completeness of H is expressed using a form of the Cauchy criterion for sequences in H : a pre-Hilbert space H is complete if every Cauchy sequence converges with respect to this norm to an element in

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1416-452: A fourth generation of fermions does not exist. Bosons are the mediators or carriers of fundamental interactions, such as electromagnetism , the weak interaction , and the strong interaction . Electromagnetism is mediated by the photon , the quanta of light . The weak interaction is mediated by the W and Z bosons . The strong interaction is mediated by the gluon , which can link quarks together to form composite particles. Due to

1534-487: A letter to Chen-Ning Yang and shortly after, Boris L. Ioffe , Lev Okun and A. P. Rudik showed that the parity violation meant that charge conjugation invariance must also be violated in weak decays. Charge violation was confirmed in the Wu experiment and in experiments performed by Valentine Telegdi and Jerome Friedman and Garwin and Lederman who observed parity non-conservation in pion and muon decay and found that C

1652-867: A long and growing list of beneficial practical applications with contributions from particle physics. Major efforts to look for physics beyond the Standard Model include the Future Circular Collider proposed for CERN and the Particle Physics Project Prioritization Panel (P5) in the US that will update the 2014 P5 study that recommended the Deep Underground Neutrino Experiment , among other experiments. Hilbert space Hilbert spaces were studied beginning in

1770-977: A non-negative integer and Ω ⊂ R , the Sobolev space H (Ω) contains L functions whose weak derivatives of order up to s are also L . The inner product in H (Ω) is ⟨ f , g ⟩ = ∫ Ω f ( x ) g ¯ ( x ) d x + ∫ Ω D f ( x ) ⋅ D g ¯ ( x ) d x + ⋯ + ∫ Ω D s f ( x ) ⋅ D s g ¯ ( x ) d x {\displaystyle \langle f,g\rangle =\int _{\Omega }f(x){\bar {g}}(x)\,\mathrm {d} x+\int _{\Omega }Df(x)\cdot D{\bar {g}}(x)\,\mathrm {d} x+\cdots +\int _{\Omega }D^{s}f(x)\cdot D^{s}{\bar {g}}(x)\,\mathrm {d} x} where

1888-408: A physically motivated point of view, von Neumann gave the first complete and axiomatic treatment of them. Von Neumann later used them in his seminal work on the foundations of quantum mechanics, and in his continued work with Eugene Wigner . The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book on quantum mechanics and the theory of groups. The significance of

2006-730: A real number x ⋅ y . If x and y are represented in Cartesian coordinates , then the dot product is defined by ( x 1 x 2 x 3 ) ⋅ ( y 1 y 2 y 3 ) = x 1 y 1 + x 2 y 2 + x 3 y 3 . {\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}\cdot {\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}\,.} The dot product satisfies

2124-579: A series of scalars, a series of vectors that converges absolutely also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as  N → ∞ . {\displaystyle {\Biggl \|}\mathbf {L} -\sum _{k=0}^{N}\mathbf {x} _{k}{\Biggr \|}\to 0\quad {\text{as }}N\to \infty \,.} This property expresses

2242-474: A special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces ) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations . They also form the basis of the theory of direct methods in the calculus of variations . For s

2360-531: A suitable sense to a square-integrable function: the missing ingredient, which ensures convergence, is completeness. The second development was the Lebesgue integral , an alternative to the Riemann integral introduced by Henri Lebesgue in 1904. The Lebesgue integral made it possible to integrate a much broader class of functions. In 1907, Frigyes Riesz and Ernst Sigismund Fischer independently proved that

2478-435: A wide range of exotic particles . All particles and their interactions observed to date can be described almost entirely by the Standard Model. Dynamics of particles are also governed by quantum mechanics ; they exhibit wave–particle duality , displaying particle-like behaviour under certain experimental conditions and wave -like behaviour in others. In more technical terms, they are described by quantum state vectors in

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2596-517: Is   0.0003   {\displaystyle \ 0.0003\ } times the maximum value of   J max = 1 6 3   ≈   0.1   . {\displaystyle \ J_{\max }={\tfrac {1}{6{\sqrt {3}}}}\ \approx \ 0.1\ .} For leptons, only an upper limit exists:   | J | < 0.03   . {\displaystyle \ |J|<0.03\ .} The reason why such

2714-636: Is countably infinite , it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable . The latter space is often in the older literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean vector space consisting of three-dimensional vectors , denoted by R , and equipped with the dot product . The dot product takes two vectors x and y , and produces

2832-509: Is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product. To say that a complex vector space H is a complex inner product space means that there is an inner product ⟨ x , y ⟩ {\displaystyle \langle x,y\rangle } associating a complex number to each pair of elements x , y {\displaystyle x,y} of H that satisfies

2950-562: Is a decomposition of z into its real and imaginary parts, then the modulus is the usual Euclidean two-dimensional length: | z | = x 2 + y 2 . {\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,.} The inner product of a pair of complex numbers z and w is the product of z with the complex conjugate of w : ⟨ z , w ⟩ = z w ¯ . {\displaystyle \langle z,w\rangle =z{\overline {w}}\,.} This

3068-428: Is a distance function means firstly that it is symmetric in x {\displaystyle x} and y , {\displaystyle y,} secondly that the distance between x {\displaystyle x} and itself is zero, and otherwise the distance between x {\displaystyle x} and y {\displaystyle y} must be positive, and lastly that

3186-425: Is a particle physics theory suggesting that systems with higher energy have a smaller number of dimensions. A third major effort in theoretical particle physics is string theory . String theorists attempt to construct a unified description of quantum mechanics and general relativity by building a theory based on small strings, and branes rather than particles. If the theory is successful, it may be considered

3304-609: Is a set, M is a σ-algebra of subsets of X , and μ is a countably additive measure on M . Let L ( X , μ ) be the space of those complex-valued measurable functions on X for which the Lebesgue integral of the square of the absolute value of the function is finite, i.e., for a function f in L ( X , μ ) , ∫ X | f | 2 d μ < ∞ , {\displaystyle \int _{X}|f|^{2}\mathrm {d} \mu <\infty \,,} and where functions are identified if and only if they differ only on

3422-535: Is also violated. Charge violation was more explicitly shown in experiments done by John Riley Holt at the University of Liverpool . Oehme then wrote a paper with Lee and Yang in which they discussed the interplay of non-invariance under P, C and T. The same result was also independently obtained by Ioffe, Okun and Rudik. Both groups also discussed possible CP violations in neutral kaon decays. Lev Landau proposed in 1957 CP-symmetry , often called just CP as

3540-486: Is called the weighted L space L w ([0, 1]) , and w is called the weight function. The inner product is defined by ⟨ f , g ⟩ = ∫ 0 1 f ( t ) g ( t ) ¯ w ( t ) d t . {\displaystyle \langle f,g\rangle =\int _{0}^{1}f(t){\overline {g(t)}}w(t)\,\mathrm {d} t\,.} The weighted space L w ([0, 1])

3658-627: Is complex-valued. The real part of ⟨ z , w ⟩ gives the usual two-dimensional Euclidean dot product . A second example is the space C whose elements are pairs of complex numbers z = ( z 1 , z 2 ) . Then an inner product of z with another such vector w = ( w 1 , w 2 ) is given by ⟨ z , w ⟩ = z 1 w 1 ¯ + z 2 w 2 ¯ . {\displaystyle \langle z,w\rangle =z_{1}{\overline {w_{1}}}+z_{2}{\overline {w_{2}}}\,.} The real part of ⟨ z , w ⟩

CP violation - Misplaced Pages Continue

3776-433: Is defined by ‖ f ‖ 2 = lim r → 1 M r ( f ) . {\displaystyle \left\|f\right\|_{2}=\lim _{r\to 1}{\sqrt {M_{r}(f)}}\,.} Hardy spaces in the disc are related to Fourier series. A function f is in H ( U ) if and only if f ( z ) = ∑ n = 0 ∞

3894-457: Is defined by: ⟨ z , w ⟩ = ∑ n = 1 ∞ z n w n ¯ , {\displaystyle \langle \mathbf {z} ,\mathbf {w} \rangle =\sum _{n=1}^{\infty }z_{n}{\overline {w_{n}}}\,,} This second series converges as a consequence of the Cauchy–Schwarz inequality and

4012-564: Is defined in the same way, except that H is a real vector space and the inner product takes real values. Such an inner product will be a bilinear map and ( H , H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,H,\langle \cdot ,\cdot \rangle )} will form a dual system . The norm is the real-valued function ‖ x ‖ = ⟨ x , x ⟩ , {\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}\,,} and

4130-458: Is different. However, consider that there are two different routes : a ⟶ 1 b {\displaystyle a{\overset {1}{\longrightarrow }}b} and a ⟶ 2 b {\displaystyle a{\overset {2}{\longrightarrow }}b} or equivalently, two unrelated intermediate states: a → 1 → b {\displaystyle a\rightarrow 1\rightarrow b} and

4248-512: Is fundamentally composed of elementary particles dates from at least the 6th century BC. In the 19th century, John Dalton , through his work on stoichiometry , concluded that each element of nature was composed of a single, unique type of particle. The word atom , after the Greek word atomos meaning "indivisible", has since then denoted the smallest particle of a chemical element , but physicists later discovered that atoms are not, in fact,

4366-590: Is identical with the Hilbert space L ([0, 1], μ ) where the measure μ of a Lebesgue-measurable set A is defined by μ ( A ) = ∫ A w ( t ) d t . {\displaystyle \mu (A)=\int _{A}w(t)\,\mathrm {d} t\,.} Weighted L spaces like this are frequently used to study orthogonal polynomials , because different families of orthogonal polynomials are orthogonal with respect to different weighting functions. Sobolev spaces , denoted by H or W , are Hilbert spaces. These are

4484-595: Is in model building where model builders develop ideas for what physics may lie beyond the Standard Model (at higher energies or smaller distances). This work is often motivated by the hierarchy problem and is constrained by existing experimental data. It may involve work on supersymmetry , alternatives to the Higgs mechanism , extra spatial dimensions (such as the Randall–Sundrum models ), Preon theory, combinations of these, or other ideas. Vanishing-dimensions theory

4602-521: Is made only from the first fermion generation. The first generation consists of up and down quarks which form protons and neutrons , and electrons and electron neutrinos . The three fundamental interactions known to be mediated by bosons are electromagnetism , the weak interaction , and the strong interaction . Quarks cannot exist on their own but form hadrons . Hadrons that contain an odd number of quarks are called baryons and those that contain an even number are called mesons . Two baryons,

4720-440: Is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg 's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras . In the 1940s, Israel Gelfand , Mark Naimark and Irving Segal gave

4838-517: Is related to both the length (or norm ) of a vector, denoted ‖ x ‖ , and to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos ⁡ θ . {\displaystyle \mathbf {x} \cdot \mathbf {y} =\left\|\mathbf {x} \right\|\left\|\mathbf {y} \right\|\,\cos \theta \,.} Multivariable calculus in Euclidean space relies on

CP violation - Misplaced Pages Continue

4956-426: Is slightly violated during certain types of weak decay . Only a weaker version of the symmetry could be preserved by physical phenomena, which was CPT symmetry . Besides C and P, there is a third operation, time reversal T , which corresponds to reversal of motion. Invariance under time reversal implies that whenever a motion is allowed by the laws of physics, the reversed motion is also an allowed one and occurs at

5074-466: Is so close to a symmetry, introduced a great puzzle. The kind of CP violation (CPV) discovered in 1964 was linked to the fact that neutral kaons can transform into their antiparticles (in which each quark is replaced with the other's antiquark) and vice versa, but such transformation does not occur with exactly the same probability in both directions; this is called indirect CP violation. Despite many searches, no other manifestation of CP violation

5192-531: Is the Jarlskog invariant :   J = c 12   c 13 2   c 23   s 12   s 13   s 23   sin ⁡ δ   ≈   0.00003   , {\displaystyle \ J=c_{12}\ c_{13}^{2}\ c_{23}\ s_{12}\ s_{13}\ s_{23}\ \sin \delta \ \approx \ 0.00003\ ,} for quarks, which

5310-527: Is the Laplacian and (1 − Δ) is understood in terms of the spectral mapping theorem . Apart from providing a workable definition of Sobolev spaces for non-integer s , this definition also has particularly desirable properties under the Fourier transform that make it ideal for the study of pseudodifferential operators . Using these methods on a compact Riemannian manifold , one can obtain for instance

5428-573: Is the study of fundamental particles and forces that constitute matter and radiation . The field also studies combinations of elementary particles up to the scale of protons and neutrons , while the study of combination of protons and neutrons is called nuclear physics . The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and bosons (force-carrying particles). There are three generations of fermions, although ordinary matter

5546-492: Is the study of these particles in radioactive processes and in particle accelerators such as the Large Hadron Collider . Theoretical particle physics is the study of these particles in the context of cosmology and quantum theory . The two are closely interrelated: the Higgs boson was postulated by theoretical particle physicists and its presence confirmed by practical experiments. The idea that all matter

5664-414: Is then the four-dimensional Euclidean dot product. This inner product is Hermitian symmetric, which means that the result of interchanging z and w is the complex conjugate: ⟨ w , z ⟩ = ⟨ z , w ⟩ ¯ . {\displaystyle \langle w,z\rangle ={\overline {\langle z,w\rangle }}\,.} A Hilbert space

5782-600: Is used to extract the parameters of the Standard Model with less uncertainty. This work probes the limits of the Standard Model and therefore expands scientific understanding of nature's building blocks. Those efforts are made challenging by the difficulty of calculating high precision quantities in quantum chromodynamics . Some theorists working in this area use the tools of perturbative quantum field theory and effective field theory , referring to themselves as phenomenologists . Others make use of lattice field theory and call themselves lattice theorists . Another major effort

5900-456: The ν μ beams. Analysis of these observations was not yet precise enough to determine the size of the CP violation, relative to that seen in quarks. In addition, another similar experiment, NOvA sees no evidence of CP violation in neutrino oscillations and is in slight tension with T2K. "Direct" CP violation is allowed in the Standard Model if a complex phase appears in

6018-805: The BaBar experiment at the Stanford Linear Accelerator Center ( SLAC ) and the Belle Experiment at the High Energy Accelerator Research Organisation ( KEK ) in Japan, observed direct CP violation in a different system, namely in decays of the B mesons . A large number of CP violation processes in B meson decays have now been discovered. Before these " B-factory " experiments, there was a logical possibility that all CP violation

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6136-676: The Cabibbo–Kobayashi–Maskawa matrix (CKM matrix) describing quark mixing, or the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix) describing neutrino mixing. A necessary condition for the appearance of the complex phase is the presence of at least three generations of fermions. If fewer generations are present, the complex phase parameter can be absorbed into redefinitions of the fermion fields. A popular rephasing invariant whose vanishing signals absence of CP violation and occurs in most CP violating amplitudes

6254-968: The Hodge decomposition , which is the basis of Hodge theory . The Hardy spaces are function spaces, arising in complex analysis and harmonic analysis , whose elements are certain holomorphic functions in a complex domain. Let U denote the unit disc in the complex plane. Then the Hardy space H ( U ) is defined as the space of holomorphic functions f on U such that the means M r ( f ) = 1 2 π ∫ 0 2 π | f ( r e i θ ) | 2 d θ {\displaystyle M_{r}(f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\left|f{\bigl (}re^{i\theta }{\bigr )}\right|^{2}\,\mathrm {d} \theta } remain bounded for r < 1 . The norm on this Hardy space

6372-708: The Lebesgue measure on the real line and unit interval, respectively, are natural domains on which to define the Fourier transform and Fourier series. In other situations, the measure may be something other than the ordinary Lebesgue measure on the real line. For instance, if w is any positive measurable function, the space of all measurable functions f on the interval [0, 1] satisfying ∫ 0 1 | f ( t ) | 2 w ( t ) d t < ∞ {\displaystyle \int _{0}^{1}{\bigl |}f(t){\bigr |}^{2}w(t)\,\mathrm {d} t<\infty }

6490-486: The Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis , in analogy with Cartesian coordinates in classical geometry. When this basis

6608-544: The atomic nuclei are baryons – the neutron is composed of two down quarks and one up quark, and the proton is composed of two up quarks and one down quark. A baryon is composed of three quarks, and a meson is composed of two quarks (one normal, one anti). Baryons and mesons are collectively called hadrons . Quarks inside hadrons are governed by the strong interaction, thus are subjected to quantum chromodynamics (color charges). The bounded quarks must have their color charge to be neutral, or "white" for analogy with mixing

6726-552: The completeness of Euclidean space: that a series that converges absolutely also converges in the ordinary sense. Hilbert spaces are often taken over the complex numbers . The complex plane denoted by C is equipped with a notion of magnitude, the complex modulus | z | , which is defined as the square root of the product of z with its complex conjugate : | z | 2 = z z ¯ . {\displaystyle |z|^{2}=z{\overline {z}}\,.} If z = x + iy

6844-417: The proton and the neutron , make up most of the mass of ordinary matter. Mesons are unstable and the longest-lived last for only a few hundredths of a microsecond . They occur after collisions between particles made of quarks, such as fast-moving protons and neutrons in cosmic rays . Mesons are also produced in cyclotrons or other particle accelerators . Particles have corresponding antiparticles with

6962-421: The quantum fields that also govern their interactions. The dominant theory explaining these fundamental particles and fields, along with their dynamics, is called the Standard Model . The reconciliation of gravity to the current particle physics theory is not solved; many theories have addressed this problem, such as loop quantum gravity , string theory and supersymmetry theory . Practical particle physics

7080-505: The same complex number. We can separate the magnitude and phase by writing M = | M |   e i θ {\displaystyle {\cal {M}}=|{\cal {M}}|\ e^{i\theta }} . If a phase term is introduced from (e.g.) the CKM matrix, denote it e i ϕ {\displaystyle e^{i\phi }} . Note that M ¯ {\displaystyle {\bar {\cal {M}}}} contains

7198-509: The time reversal symmetry violation without any assumption of CPT theorem was done in 1998 by two groups, CPLEAR and KTeV collaborations, at CERN and Fermilab , respectively. Already in 1970 Klaus Schubert observed T violation independent of assuming CPT symmetry by using the Bell–Steinberger unitarity relation. The idea behind parity symmetry was that the equations of particle physics are invariant under mirror inversion. This led to

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7316-824: The triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the other two legs: d ( x , z ) ≤ d ( x , y ) + d ( y , z ) . {\displaystyle d(x,z)\leq d(x,y)+d(y,z)\,.} This last property is ultimately a consequence of the more fundamental Cauchy–Schwarz inequality , which asserts | ⟨ x , y ⟩ | ≤ ‖ x ‖ ‖ y ‖ {\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|\|y\|} with equality if and only if x {\displaystyle x} and y {\displaystyle y} are linearly dependent . With

7434-457: The 1950s and 1960s, a bewildering variety of particles was found in collisions of particles from beams of increasingly high energy. It was referred to informally as the " particle zoo ". Important discoveries such as the CP violation by James Cronin and Val Fitch brought new questions to matter-antimatter imbalance . After the formulation of the Standard Model during the 1970s, physicists clarified

7552-409: The 1980 Nobel Prize. This discovery showed that weak interactions violate not only the charge-conjugation symmetry C between particles and antiparticles and the P or parity symmetry, but also their combination. The discovery shocked particle physics and opened the door to questions still at the core of particle physics and of cosmology today. The lack of an exact CP-symmetry, but also the fact that it

7670-653: The ability to compute limits , and to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n {\displaystyle \sum _{n=0}^{\infty }\mathbf {x} _{n}} consisting of vectors in R is absolutely convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞ . {\displaystyle \sum _{k=0}^{\infty }\|\mathbf {x} _{k}\|<\infty \,.} Just as with

7788-571: The aforementioned color confinement, gluons are never observed independently. The Higgs boson gives mass to the W and Z bosons via the Higgs mechanism – the gluon and photon are expected to be massless . All bosons have an integer quantum spin (0 and 1) and can have the same quantum state . Most aforementioned particles have corresponding antiparticles , which compose antimatter . Normal particles have positive lepton or baryon number , and antiparticles have these numbers negative. Most properties of corresponding antiparticles and particles are

7906-499: The concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics . In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators , and measurements are orthogonal projections . The relation between quantum mechanical symmetries and unitary operators provided an impetus for

8024-962: The conjugate matrix to M {\displaystyle {\cal {M}}} , so it picks up a phase term e − i ϕ {\displaystyle e^{-i\phi }} . Now the formula becomes: M = | M |   e i θ   e + i ϕ M ¯ = | M |   e i θ   e − i ϕ {\displaystyle {\begin{aligned}{\cal {M}}&=|{\cal {M}}|\ e^{i\theta }\ e^{+i\phi }\\{\bar {\cal {M}}}&=|{\cal {M}}|\ e^{i\theta }\ e^{-i\phi }\end{aligned}}} Physically measurable reaction rates are proportional to   | M | 2 {\displaystyle \ |{\cal {M}}|^{2}} , thus so far nothing

8142-597: The constituents of all matter . Finally, the Standard Model also predicted the existence of a type of boson known as the Higgs boson . On 4 July 2012, physicists with the Large Hadron Collider at CERN announced they had found a new particle that behaves similarly to what is expected from the Higgs boson. The Standard Model, as currently formulated, has 61 elementary particles. Those elementary particles can combine to form composite particles, accounting for

8260-440: The convergence of the previous series. Completeness of the space holds provided that whenever a series of elements from l converges absolutely (in norm), then it converges to an element of l . The proof is basic in mathematical analysis , and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a finite-dimensional Euclidean space). Prior to

8378-517: The decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch . It plays an important role both in the attempts of cosmology to explain the dominance of matter over antimatter in the present universe , and in the study of weak interactions in particle physics. Until the 1950s, parity conservation was believed to be one of

8496-551: The development of Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists . In particular, the idea of an abstract linear space (vector space) had gained some traction towards the end of the 19th century: this is a space whose elements can be added together and multiplied by scalars (such as real or complex numbers ) without necessarily identifying these elements with "geometric" vectors , such as position and momentum vectors in physical systems. Other objects studied by mathematicians at

8614-479: The development of the unitary representation theory of groups , initiated in the 1928 work of Hermann Weyl. On the other hand, in the early 1930s it became clear that classical mechanics can be described in terms of Hilbert space ( Koopman–von Neumann classical mechanics ) and that certain properties of classical dynamical systems can be analyzed using Hilbert space techniques in the framework of ergodic theory . The algebra of observables in quantum mechanics

8732-471: The distance d {\displaystyle d} between two points x , y {\displaystyle x,y} in H is defined in terms of the norm by d ( x , y ) = ‖ x − y ‖ = ⟨ x − y , x − y ⟩ . {\displaystyle d(x,y)=\|x-y\|={\sqrt {\langle x-y,x-y\rangle }}\,.} That this function

8850-846: The dot indicates the dot product in the Euclidean space of partial derivatives of each order. Sobolev spaces can also be defined when s is not an integer. Sobolev spaces are also studied from the point of view of spectral theory, relying more specifically on the Hilbert space structure. If Ω is a suitable domain, then one can define the Sobolev space H (Ω) as the space of Bessel potentials ; roughly, H s ( Ω ) = { ( 1 − Δ ) − s / 2 f | f ∈ L 2 ( Ω ) } . {\displaystyle H^{s}(\Omega )=\left\{(1-\Delta )^{-s/2}f\mathrel {\Big |} f\in L^{2}(\Omega )\right\}\,.} Here Δ

8968-460: The early 20th century. For example, the Riesz representation theorem was independently established by Maurice Fréchet and Frigyes Riesz in 1907. John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators . Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from

9086-485: The eigenvalues are given by m 1 2 = A − B x y − C x 2 + y 2 + x 2 y 2 x y , {\displaystyle \mathbf {m_{1}} ^{2}=\mathbf {A} -\mathbf {B} {x \over y}-\mathbf {C} {{\sqrt {x^{2}+y^{2}+x^{2}y^{2}}} \over xy},} Particle physics Particle physics or high-energy physics

9204-1156: The fermion mass matrices M u {\displaystyle M_{u}} and M d {\displaystyle M_{d}} , respectively. Thus, there are two necessary conditions for getting a complex CKM matrix: For a standard model with three fermion generations, the most general non-Hermitian pattern of its mass matrices can be given by M = [ A 1 + i D 1 B 1 + i C 1 B 2 + i C 2 B 4 + i C 4 A 2 + i D 2 B 3 + i C 3 B 5 + i C 5 B 6 + i C 6 A 3 + i D 3 ] . {\displaystyle M={\begin{bmatrix}A_{1}+iD_{1}&B_{1}+iC_{1}&B_{2}+iC_{2}\\B_{4}+iC_{4}&A_{2}+iD_{2}&B_{3}+iC_{3}\\B_{5}+iC_{5}&B_{6}+iC_{6}&A_{3}+iD_{3}\end{bmatrix}}.} This M matrix contains 9 elements and 18 parameters, 9 from

9322-403: The first decade of the 20th century by David Hilbert , Erhard Schmidt , and Frigyes Riesz . They are indispensable tools in the theories of partial differential equations , quantum mechanics , Fourier analysis (which includes applications to signal processing and heat transfer ), and ergodic theory (which forms the mathematical underpinning of thermodynamics ). John von Neumann coined

9440-478: The first experimental deviations from the Standard Model, since neutrinos do not have mass in the Standard Model. Modern particle physics research is focused on subatomic particles , including atomic constituents, such as electrons , protons , and neutrons (protons and neutrons are composite particles called baryons , made of quarks ), that are produced by radioactive and scattering processes; such particles are photons , neutrinos , and muons , as well as

9558-442: The first time. In this experiment, beams of muon neutrinos ( ν μ ) and muon antineutrinos ( ν μ ) were alternately produced by an accelerator . By the time they got to the detector, a significantly higher proportion of electron neutrinos ( ν e ) was observed from the ν μ beams, than electron antineutrinos ( ν e ) were from

9676-628: The following properties: It follows from properties 1 and 2 that a complex inner product is antilinear , also called conjugate linear , in its second argument, meaning that ⟨ x , a y 1 + b y 2 ⟩ = a ¯ ⟨ x , y 1 ⟩ + b ¯ ⟨ x , y 2 ⟩ . {\displaystyle \langle x,ay_{1}+by_{2}\rangle ={\bar {a}}\langle x,y_{1}\rangle +{\bar {b}}\langle x,y_{2}\rangle \,.} A real inner product space

9794-418: The fundamental geometric conservation laws (along with conservation of energy and conservation of momentum ). After the discovery of parity violation in 1956, CP-symmetry was proposed to restore order. However, while the strong interaction and electromagnetic interaction are experimentally found to be invariant under the combined CP transformation operation, further experiments showed that this symmetry

9912-424: The fundamental particles of nature, but are conglomerates of even smaller particles, such as the electron . The early 20th century explorations of nuclear physics and quantum physics led to proofs of nuclear fission in 1939 by Lise Meitner (based on experiments by Otto Hahn ), and nuclear fusion by Hans Bethe in that same year; both discoveries also led to the development of nuclear weapons . Throughout

10030-538: The gravitational interaction, but it has not been detected or completely reconciled with current theories. Many other hypothetical particles have been proposed to address the limitations of the Standard Model. Notably, supersymmetric particles aim to solve the hierarchy problem , axions address the strong CP problem , and various other particles are proposed to explain the origins of dark matter and dark energy . The world's major particle physics laboratories are: Theoretical particle physics attempts to develop

10148-424: The hundreds of other species of particles that have been discovered since the 1960s. The Standard Model has been found to agree with almost all the experimental tests conducted to date. However, most particle physicists believe that it is an incomplete description of nature and that a more fundamental theory awaits discovery (See Theory of Everything ). In recent years, measurements of neutrino mass have provided

10266-433: The interactions between the quarks store energy which can convert to other particles when the quarks are far apart enough, quarks cannot be observed independently. This is called color confinement . There are three known generations of quarks (up and down, strange and charm , top and bottom ) and leptons (electron and its neutrino, muon and its neutrino , tau and its neutrino ), with strong indirect evidence that

10384-447: The introduction of Hilbert spaces. The first of these was the observation, which arose during David Hilbert and Erhard Schmidt 's study of integral equations , that two square-integrable real-valued functions f and g on an interval [ a , b ] have an inner product that has many of the familiar properties of the Euclidean dot product. In particular, the idea of an orthogonal family of functions has meaning. Schmidt exploited

10502-456: The mid-1970s after experimental confirmation of the existence of quarks . It describes the strong , weak , and electromagnetic fundamental interactions , using mediating gauge bosons . The species of gauge bosons are eight gluons , W , W and Z bosons , and the photon . The Standard Model also contains 24 fundamental fermions (12 particles and their associated anti-particles), which are

10620-497: The models, theoretical framework, and mathematical tools to understand current experiments and make predictions for future experiments (see also theoretical physics ). There are several major interrelated efforts being made in theoretical particle physics today. One important branch attempts to better understand the Standard Model and its tests. Theorists make quantitative predictions of observables at collider and astronomical experiments, which along with experimental measurements

10738-798: The most general ones. The perfect way to solve the CPV problem in the standard model is to diagonalize such matrices analytically and to achieve a U matrix which applies to both. Unfortunately, even though the M 2 {\displaystyle \mathbf {M^{2}} } matrix has only 9 parameters, it is still too complicated to be diagonalized directly. Thus, an assumption M 2 R ⋅ M 2 † I + M 2 I ⋅ M 2 † R = 0 {\displaystyle \mathbf {M^{2}} _{R}\cdot \mathbf {M^{2\dagger }} _{I}+\mathbf {M^{2}} _{I}\cdot \mathbf {M^{2\dagger }} _{R}=0}

10856-426: The origin of the particle zoo. The large number of particles was explained as combinations of a (relatively) small number of more fundamental particles and framed in the context of quantum field theories . This reclassification marked the beginning of modern particle physics. The current state of the classification of all elementary particles is explained by the Standard Model , which gained widespread acceptance in

10974-2018: The parameter number from 9 to 5 and the reduced M 2 {\displaystyle \mathbf {M^{2}} } matrix can be given by M 2 = [ A + B ( x y − x y ) y B x B y B A + B ( y x − x y ) B x B B A ] + i [ 0 C y − C x − C y 0 C C x − C 0 ] ≡ M 2 R + i M 2 I , {\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A} +\mathbf {B} (xy-{x \over y})&y\mathbf {B} &x\mathbf {B} \\y\mathbf {B} &\mathbf {A} +\mathbf {B} ({y \over x}-{x \over y})&\mathbf {B} \\x\mathbf {B} &\mathbf {B} &\mathbf {A} \end{bmatrix}}+i{\begin{bmatrix}0&{\mathbf {C} \over y}&-{\mathbf {C} \over x}\\-{\mathbf {C} \over y}&0&\mathbf {C} \\{\mathbf {C} \over x}&-\mathbf {C} &0\end{bmatrix}}\equiv \mathbf {M^{2}} _{R}+i\mathbf {M^{2}} _{I},} where A ≡ A 3 , B ≡ B 3 , C ≡ C 3 , x ≡ B 2 / B 3 , {\displaystyle \mathbf {A} \equiv \mathbf {A_{3}} ,\mathbf {B} \equiv \mathbf {B_{3}} ,\mathbf {C} \equiv \mathbf {C_{3}} ,x\equiv \mathbf {B_{2}/B_{3}} ,} and y ≡ B 1 / B 3 {\displaystyle y\equiv \mathbf {B_{1}/B_{3}} } . Diagonalizing M 2 {\displaystyle \mathbf {M^{2}} } analytically,

11092-483: The photon or gluon, have no antiparticles. Quarks and gluons additionally have color charges, which influences the strong interaction. Quark's color charges are called red, green and blue (though the particle itself have no physical color), and in antiquarks are called antired, antigreen and antiblue. The gluon can have eight color charges , which are the result of quarks' interactions to form composite particles (gauge symmetry SU(3) ). The neutrons and protons in

11210-413: The prediction that the mirror image of a reaction (such as a chemical reaction or radioactive decay ) occurs at the same rate as the original reaction. However, in 1956 a careful critical review of the existing experimental data by theoretical physicists Tsung-Dao Lee and Chen-Ning Yang revealed that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it

11328-426: The primary colors . More exotic hadrons can have other types, arrangement or number of quarks ( tetraquark , pentaquark ). An atom is made from protons, neutrons and electrons. By modifying the particles inside a normal atom, exotic atoms can be formed. A simple example would be the hydrogen-4.1 , which has one of its electrons replaced with a muon. The graviton is a hypothetical particle that can mediate

11446-573: The processes   a → b   {\displaystyle \ a\rightarrow b\ } and the corresponding antiparticle process   a ¯ → b ¯   , {\displaystyle \ {\bar {a}}\rightarrow {\bar {b}}\ ,} and denote their amplitudes M {\displaystyle {\cal {M}}} and M ¯ {\displaystyle {\bar {\cal {M}}}} respectively. Before CP violation, these terms must be

11564-406: The properties An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product . A vector space equipped with such an inner product is known as a (real) inner product space . Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it

11682-1248: The real coefficients and 9 from the imaginary coefficients. Obviously, a 3x3 matrix with 18 parameters is too difficult to diagonalize analytically. However, a naturally Hermitian M 2 = M ⋅ M † {\displaystyle \mathbf {M^{2}} =M\cdot M^{\dagger }} can be given by M 2 = [ A 1 B 1 + i C 1 B 2 + i C 2 B 1 − i C 1 A 2 B 3 + i C 3 B 2 − i C 2 B 3 − i C 3 A 3 ] , {\displaystyle \mathbf {M^{2}} ={\begin{bmatrix}\mathbf {A_{1}} &\mathbf {B_{1}} +i\mathbf {C_{1}} &\mathbf {B_{2}} +i\mathbf {C_{2}} \\\mathbf {B_{1}} -i\mathbf {C_{1}} &\mathbf {A_{2}} &\mathbf {B_{3}} +i\mathbf {C_{3}} \\\mathbf {B_{2}} -i\mathbf {C_{2}} &\mathbf {B_{3}} -i\mathbf {C_{3}} &\mathbf {A_{3}} \end{bmatrix}},} and it has

11800-412: The same mass but with opposite electric charges . For example, the antiparticle of the electron is the positron . The electron has a negative electric charge, the positron has a positive charge. These antiparticles can theoretically form a corresponding form of matter called antimatter . Some particles, such as the photon , are their own antiparticle. These elementary particles are excitations of

11918-430: The same effect as diagonalizing an M {\displaystyle M} matrix with 18 parameters. Therefore, diagonalizing the M 2 {\displaystyle \mathbf {M^{2}} } matrix is certainly the most reasonable choice. The M {\displaystyle M} and M 2 {\displaystyle \mathbf {M^{2}} } matrix patterns given above are

12036-526: The same measurement using the full 3.0 fb Run 1 sample was consistent with CP-symmetry. In 2013 LHCb announced discovery of CP violation in strange B meson decays. In March 2019, LHCb announced discovery of CP violation in charmed D 0 {\displaystyle D^{0}} decays with a deviation from zero of 5.3 standard deviations. In 2020, the T2K Collaboration reported some indications of CP violation in leptons for

12154-595: The same rate forwards and backwards. The combination of CPT is thought to constitute an exact symmetry of all types of fundamental interactions. Because of the long-held CPT symmetry theorem, provided that it is valid, a violation of the CP-symmetry is equivalent to a violation of the T-symmetry. In this theorem, regarded as one of the basic principles of quantum field theory , charge conjugation, parity, and time reversal are applied together. Direct observation of

12272-3739: The same unitary transformation matrix U with M. Besides, parameters in M 2 {\displaystyle \mathbf {M^{2}} } are correlated to those in M directly in the ways shown below A 1 = A 1 2 + D 1 2 + B 1 2 + C 1 2 + B 2 2 + C 2 2 , A 2 = A 2 2 + D 2 2 + B 3 2 + C 3 2 + B 4 2 + C 4 2 , A 3 = A 3 2 + D 3 2 + B 5 2 + C 5 2 + B 6 2 + C 6 2 , B 1 = A 1 B 4 + D 1 C 4 + B 1 A 2 + C 1 D 2 + B 2 B 3 + C 2 C 3 , B 2 = A 1 B 5 + D 1 C 5 + B 1 B 6 + C 1 C 6 + B 2 A 3 + C 2 D 3 , B 3 = B 4 B 5 + C 4 C 5 + B 6 A 2 + C 6 D 2 + A 3 B 3 + D 3 C 3 , C 1 = D 1 B 4 − A 1 C 4 + A 2 C 1 − B 1 D 2 + B 3 C 2 − B 2 C 3 , C 2 = D 1 B 5 − A 1 C 5 + B 6 C 1 − B 1 C 6 + A 3 C 2 − B 2 D 3 , C 3 = C 4 B 5 − B 4 C 5 + D 2 B 6 − A 2 C 6 + A 3 C 3 − B 3 D 3 . {\displaystyle {\begin{aligned}\mathbf {A_{1}} &=A_{1}^{2}+D_{1}^{2}+B_{1}^{2}+C_{1}^{2}+B_{2}^{2}+C_{2}^{2},\\\mathbf {A_{2}} &=A_{2}^{2}+D_{2}^{2}+B_{3}^{2}+C_{3}^{2}+B_{4}^{2}+C_{4}^{2},\\\mathbf {A_{3}} &=A_{3}^{2}+D_{3}^{2}+B_{5}^{2}+C_{5}^{2}+B_{6}^{2}+C_{6}^{2},\\\mathbf {B_{1}} &=A_{1}B_{4}+D_{1}C_{4}+B_{1}A_{2}+C_{1}D_{2}+B_{2}B_{3}+C_{2}C_{3},\\\mathbf {B_{2}} &=A_{1}B_{5}+D_{1}C_{5}+B_{1}B_{6}+C_{1}C_{6}+B_{2}A_{3}+C_{2}D_{3},\\\mathbf {B_{3}} &=B_{4}B_{5}+C_{4}C_{5}+B_{6}A_{2}+C_{6}D_{2}+A_{3}B_{3}+D_{3}C_{3},\\\mathbf {C_{1}} &=D_{1}B_{4}-A_{1}C_{4}+A_{2}C_{1}-B_{1}D_{2}+B_{3}C_{2}-B_{2}C_{3},\\\mathbf {C_{2}} &=D_{1}B_{5}-A_{1}C_{5}+B_{6}C_{1}-B_{1}C_{6}+A_{3}C_{2}-B_{2}D_{3},\\\mathbf {C_{3}} &=C_{4}B_{5}-B_{4}C_{5}+D_{2}B_{6}-A_{2}C_{6}+A_{3}C_{3}-B_{3}D_{3}.\end{aligned}}} That means if we diagonalize an M 2 {\displaystyle \mathbf {M^{2}} } matrix with 9 parameters, it has

12390-444: The same, with a few gets reversed; the electron's antiparticle, positron, has an opposite charge. To differentiate between antiparticles and particles, a plus or negative sign is added in superscript . For example, the electron and the positron are denoted e and e . When a particle and an antiparticle interact with each other, they are annihilated and convert to other particles. Some particles, such as

12508-657: The second form (conjugation of the first element) is commonly found in the theoretical physics literature. For f and g in L , the integral exists because of the Cauchy–Schwarz inequality, and defines an inner product on the space. Equipped with this inner product, L is in fact complete. The Lebesgue integral is essential to ensure completeness: on domains of real numbers, for instance, not enough functions are Riemann integrable . The Lebesgue spaces appear in many natural settings. The spaces L ( R ) and L ([0,1]) of square-integrable functions with respect to

12626-400: The series converges in H , in the sense that the partial sums converge to an element of H . As a complete normed space, Hilbert spaces are by definition also Banach spaces . As such they are topological vector spaces , in which topological notions like the openness and closedness of subsets are well defined . Of special importance is the notion of a closed linear subspace of

12744-576: The similarity of this inner product with the usual dot product to prove an analog of the spectral decomposition for an operator of the form where K is a continuous function symmetric in x and y . The resulting eigenfunction expansion expresses the function K as a series of the form where the functions φ n are orthogonal in the sense that ⟨ φ n , φ m ⟩ = 0 for all n ≠ m . The individual terms in this series are sometimes referred to as elementary product solutions. However, there are eigenfunction expansions that fail to converge in

12862-522: The space L of square Lebesgue-integrable functions is a complete metric space . As a consequence of the interplay between geometry and completeness, the 19th century results of Joseph Fourier , Friedrich Bessel and Marc-Antoine Parseval on trigonometric series easily carried over to these more general spaces, resulting in a geometrical and analytical apparatus now usually known as the Riesz–Fischer theorem . Further basic results were proved in

12980-496: The space. Completeness can be characterized by the following equivalent condition: if a series of vectors ∑ k = 0 ∞ u k {\displaystyle \sum _{k=0}^{\infty }u_{k}} converges absolutely in the sense that ∑ k = 0 ∞ ‖ u k ‖ < ∞ , {\displaystyle \sum _{k=0}^{\infty }\|u_{k}\|<\infty \,,} then

13098-467: The symmetry of a quantum mechanical system can be restored if another approximate symmetry S can be found such that the combined symmetry PS remains unbroken. This rather subtle point about the structure of Hilbert space was realized shortly after the discovery of P violation, and it was proposed that charge conjugation, C , which transforms a particle into its antiparticle , was the suitable symmetry to restore order. In 1956 Reinhard Oehme in

13216-551: The term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis . Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions , spaces of sequences , Sobolev spaces consisting of generalized functions , and Hardy spaces of holomorphic functions . Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of

13334-677: The true symmetry between matter and antimatter. CP-symmetry is the product of two transformations : C for charge conjugation and P for parity. In other words, a process in which all particles are exchanged with their antiparticles was assumed to be equivalent to the mirror image of the original process and so the combined CP-symmetry would be conserved in the weak interaction. In 1962, a group of experimentalists at Dubna , on Okun's insistence, unsuccessfully searched for CP-violating kaon decay. In 1964, James Cronin , Val Fitch and coworkers provided clear evidence from kaon decay that CP-symmetry could be broken. (cf. also Ref. ). This work won them

13452-425: The turn of the 20th century, in particular spaces of sequences (including series ) and spaces of functions, can naturally be thought of as linear spaces. Functions, for instance, can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors. In the first decade of the 20th century, parallel developments led to

13570-460: Was confined to kaon physics. However, this raised the question of why CP violation did not extend to the strong force, and furthermore, why this was not predicted by the unextended Standard Model , despite the model's accuracy for "normal" phenomena. In 2011, a hint of CP violation in decays of neutral D mesons was reported by the LHCb experiment at CERN using 0.6 fb of Run 1 data. However,

13688-551: Was discovered until the 1990s, when the NA31 experiment at CERN suggested evidence for CP violation in the decay process of the very same neutral kaons ( direct CP violation). The observation was somewhat controversial, and final proof for it came in 1999 from the KTeV experiment at Fermilab and the NA48 experiment at CERN . Starting in 2001, a new generation of experiments, including

13806-402: Was employed to simplify the pattern, where M 2 R {\displaystyle \mathbf {M^{2}} _{R}} is the real part of M 2 {\displaystyle \mathbf {M^{2}} } and M 2 I {\displaystyle \mathbf {M^{2}} _{I}} is the imaginary part. Such an assumption could further reduce

13924-577: Was untested in the weak interaction. They proposed several possible direct experimental tests. The first test based on beta decay of cobalt-60 nuclei was carried out in 1956 by a group led by Chien-Shiung Wu , and demonstrated conclusively that weak interactions violate the P-symmetry or, as the analogy goes, some reactions did not occur as often as their mirror image. However, parity symmetry still appears to be valid for all reactions involving electromagnetism and strong interactions . Overall,

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