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167-490: These discrete symmetries, C, P and T, are symmetries of the equations that describe the known fundamental forces of nature: electromagnetism , gravity , the strong and the weak interactions . Verifying whether some given mathematical equation correctly models nature requires giving physical interpretation not only to continuous symmetries , such as motion in time, but also to its discrete symmetries , and then determining whether nature adheres to these symmetries. Unlike

334-406: A 1 e 1 , … , a k e k {\displaystyle a_{1}\mathbf {e} _{1},\ldots ,a_{k}\mathbf {e} _{k}} is a basis of G , for some nonzero integers a 1 , … , a k {\displaystyle a_{1},\ldots ,a_{k}} . For details, see Free abelian group § Subgroups . In

501-506: A i , j v i = ∑ i = 1 n ( ∑ j = 1 n a i , j y j ) v i . {\displaystyle \mathbf {x} =\sum _{j=1}^{n}y_{j}\mathbf {w} _{j}=\sum _{j=1}^{n}y_{j}\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}=\sum _{i=1}^{n}{\biggl (}\sum _{j=1}^{n}a_{i,j}y_{j}{\biggr )}\mathbf {v} _{i}.} The change-of-basis formula results then from

668-666: A k , b k . But many square-integrable functions cannot be represented as finite linear combinations of these basis functions, which therefore do not comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are typically not useful, whereas orthonormal bases of these spaces are essential in Fourier analysis . The geometric notions of an affine space , projective space , convex set , and cone have related notions of basis . An affine basis for an n -dimensional affine space

835-410: A , b ) + ( c , d ) = ( a + c , b + d ) {\displaystyle (a,b)+(c,d)=(a+c,b+d)} and scalar multiplication λ ( a , b ) = ( λ a , λ b ) , {\displaystyle \lambda (a,b)=(\lambda a,\lambda b),} where λ {\displaystyle \lambda }

1002-508: A central force without a communicating medium. Thus Newton's theory violated the tradition, going back to Descartes , that there should be no action at a distance . Conversely, during the 1820s, when explaining magnetism, Michael Faraday inferred a field filling space and transmitting that force. Faraday conjectured that ultimately, all forces unified into one. In 1873, James Clerk Maxwell unified electricity and magnetism as effects of an electromagnetic field whose third consequence

1169-577: A Clifford bundle and a spin manifold . At the end of this construction, one obtains a system that is remarkably familiar, if one is already acquainted with Dirac spinors and the Dirac equation. Several analogies pass through to this general case. First, the spinors are the Weyl spinors , and they come in complex-conjugate pairs. They are naturally anti-commuting (this follows from the Clifford algebra), which

1336-438: A Hilbert basis (linear programming) . For a probability distribution in R with a probability density function , such as the equidistribution in an n -dimensional ball with respect to Lebesgue measure, it can be shown that n randomly and independently chosen vectors will form a basis with probability one , which is due to the fact that n linearly dependent vectors x 1 , ..., x n in R should satisfy

1503-414: A fifth force might exist, but these hypotheses remain speculative. Each of the known fundamental interactions can be described mathematically as a field . The gravitational force is attributed to the curvature of spacetime , described by Einstein's general theory of relativity . The other three are discrete quantum fields , and their interactions are mediated by elementary particles described by

1670-469: A linearly independent set L of n elements of V , one may replace n well-chosen elements of S by the elements of L to get a spanning set containing L , having its other elements in S , and having the same number of elements as S . Most properties resulting from the Steinitz exchange lemma remain true when there is no finite spanning set, but their proofs in the infinite case generally require

1837-406: A set B of vectors in a vector space V is called a basis ( pl. : bases ) if every element of V may be written in a unique way as a finite linear combination of elements of B . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B . The elements of a basis are called basis vectors . Equivalently, a set B

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2004-462: A tangent bundle , a cotangent bundle and a metric that ties the two together. There are several interesting things one can do, when presented with this situation. One is that the smooth structure allows differential equations to be posed on the manifold; the tangent and cotangent spaces provide enough structure to perform calculus on manifolds . Of key interest is the Laplacian , and, with

2171-539: A virtual Higgs boson, yielding classical potentials of the form with Higgs mass 125.18 GeV . Because the reduced Compton wavelength of the Higgs boson is so small ( 1.576 × 10  m , comparable to the W and Z bosons ), this potential has an effective range of a few attometers . Between two electrons, it begins roughly 10 times weaker than the weak interaction , and grows exponentially weaker at non-zero distances. Coordinate frame In mathematics ,

2338-800: A 4×4 matrix representation! More precisely, there is no complex 4×4 matrix that can take a complex number to its complex conjugate; this inversion would require an 8×8 real matrix. The physical interpretation of complex conjugation as charge conjugation becomes clear when considering the complex conjugation of scalar fields, described in a subsequent section below. The projectors onto the chiral eigenstates can be written as P L = ( 1 − γ 5 ) / 2 {\displaystyle P_{\text{L}}=\left(1-\gamma _{5}\right)/2} and P R = ( 1 + γ 5 ) / 2 , {\displaystyle P_{\text{R}}=\left(1+\gamma _{5}\right)/2,} and so

2505-660: A backbone, M-theory . Theories beyond the Standard Model remain highly speculative, lacking great experimental support. In the conceptual model of fundamental interactions, matter consists of fermions , which carry properties called charges and spin ± 1 ⁄ 2 (intrinsic angular momentum ± ħ ⁄ 2 , where ħ is the reduced Planck constant ). They attract or repel each other by exchanging bosons . The interaction of any pair of fermions in perturbation theory can then be modelled thus: The exchange of bosons always carries energy and momentum between

2672-448: A basis of R . More generally, if F is a field , the set F n {\displaystyle F^{n}} of n -tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let e i = ( 0 , … , 0 , 1 , 0 , … , 0 ) {\displaystyle \mathbf {e} _{i}=(0,\ldots ,0,1,0,\ldots ,0)} be

2839-580: A basis of V . By definition of a basis, every v in V may be written, in a unique way, as v = λ 1 b 1 + ⋯ + λ n b n , {\displaystyle \mathbf {v} =\lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n},} where the coefficients λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are scalars (that is, elements of F ), which are called

3006-420: A change of local coordinate frames on the circle. For U(1), this is just the statement that the system is invariant under multiplication by a phase factor e i ϕ ( x ) {\displaystyle e^{i\phi (x)}} that depends on the (space-time) coordinate x . {\displaystyle x.} In this geometric setting, charge conjugation can be understood as

3173-404: A charge, and exchange virtual particles ( gauge bosons ), which are the interaction carriers or force mediators. For example, photons mediate the interaction of electric charges , and gluons mediate the interaction of color charges . The full theory includes perturbations beyond simply fermions exchanging bosons; these additional perturbations can involve bosons that exchange fermions, as well as

3340-448: A common theoretical framework with the other three forces. Some theories, notably string theory , seek both QG and GUT within one framework, unifying all four fundamental interactions along with mass generation within a theory of everything (ToE). In his 1687 theory, Isaac Newton postulated space as an infinite and unalterable physical structure existing before, within, and around all objects while their states and relations unfold at

3507-410: A complex-number-valued structure to be coupled to the electromagnetic field, provided that this coupling is done in a gauge-invariant way. Gauge symmetry, in this geometric setting, is a statement that, as one moves around on the circle, the coupled object must also transform in a "circular way", tracking in a corresponding fashion. More formally, one says that the equations must be gauge invariant under

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3674-400: A constant pace everywhere, thus absolute space and time . Inferring that all objects bearing mass approach at a constant rate, but collide by impact proportional to their masses, Newton inferred that matter exhibits an attractive force. His law of universal gravitation implied there to be instant interaction among all objects. As conventionally interpreted, Newton's theory of motion modelled

3841-538: A constant term, what amounts to the Klein–Gordon operator. Cotangent bundles, by their basic construction, are always symplectic manifolds . Symplectic manifolds have canonical coordinates x , p {\displaystyle x,p} interpreted as position and momentum, obeying canonical commutation relations . This provides the core infrastructure to extend duality, and thus charge conjugation, to this general setting. A second interesting thing one can do

4008-456: A coordinate frame. Put another way, a spinor field is a local section of the spinor bundle, and Lorentz boosts and rotations correspond to movements along the fibers of the corresponding frame bundle (again, just a choice of local coordinate frame). Examined in this way, this extra phase freedom can be interpreted as the phase arising from the electromagnetic field. For the Majorana spinors ,

4175-558: A field set to special relativity , altogether relativistic quantum field theory (QFT). Force particles, called gauge bosons — force carriers or messenger particles of underlying fields—interact with matter particles, called fermions . Everyday matter is atoms, composed of three fermion types: up-quarks and down-quarks constituting, as well as electrons orbiting, the atom's nucleus. Atoms interact, form molecules , and manifest further properties through electromagnetic interactions among their electrons absorbing and emitting photons,

4342-388: A free abelian group is a free abelian group, and, if G is a subgroup of a finitely generated free abelian group H (that is an abelian group that has a finite basis), then there is a basis e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} of H and an integer 0 ≤ k ≤ n such that

4509-413: A geometric interpretation. It has been noted that, for massive Dirac spinors, the "arbitrary" phase factor   η c   {\displaystyle \ \eta _{c}\ } may depend on both the momentum, and the helicity (but not the chirality). This can be interpreted as saying that this phase may vary along the fiber of the spinor bundle , depending on the local choice of

4676-409: A knot, one finally has the concept of transposition , in that elements of the Clifford algebra can be written in reversed (transposed) order. The net result is that not only do the conventional physics ideas of fields pass over to the general Riemannian setting, but also the ideas of the discrete symmetries. There are two ways to react to this. One is to treat it as an interesting curiosity. The other

4843-479: A linear operator, one may consider its eigenstates. The Majorana condition singles out one such: C ψ = ψ . {\displaystyle {\mathsf {C}}\psi =\psi .} There are, however, two such eigenstates: C ψ ( ± ) = ± ψ ( ± ) . {\displaystyle {\mathsf {C}}\psi ^{(\pm )}=\pm \psi ^{(\pm )}.} Continuing in

5010-505: A linear potential, a constant attractive force. In this way, the mathematical theory of QCD not only explains how quarks interact over short distances but also the string-like behavior, discovered by Chew and Frautschi, which they manifest over longer distances. Conventionally, the Higgs interaction is not counted among the four fundamental forces. Nonetheless, although not a gauge interaction nor generated by any diffeomorphism symmetry,

5177-406: A meter apart, the electrons in one of the jugs repel those in the other jug with a force of This force is many times larger than the weight of the planet Earth. The atomic nuclei in one jug also repel those in the other with the same force. However, these repulsive forces are canceled by the attraction of the electrons in jug A with the nuclei in jug B and the attraction of the nuclei in jug A with

C-symmetry - Misplaced Pages Continue

5344-484: A minimal understanding of what quantum field theory is. In (vastly) simplified terms, it is a technique for performing calculations to obtain solutions for a system of coupled differential equations via perturbation theory . A key ingredient to this process is the quantum field , one for each of the (free, uncoupled) differential equations in the system. A quantum field is conventionally written as where p → {\displaystyle {\vec {p}}}

5511-420: A net electric charge of zero. Nothing "cancels" gravity, since it is only attractive, unlike electric forces which can be attractive or repulsive. On the other hand, all objects having mass are subject to the gravitational force, which only attracts. Therefore, only gravitation matters on the large-scale structure of the universe. The long range of gravitation makes it responsible for such large-scale phenomena as

5678-472: A piece F = d A {\displaystyle F=dA} with A {\displaystyle A} arising from that part of the connection associated with the U ( 1 ) {\displaystyle U(1)} piece. This is entirely analogous to what happens when one squares the ordinary Dirac equation in ordinary Minkowski spacetime. A second hint is that this U ( 1 ) {\displaystyle U(1)} piece

5845-559: A plane-wave state ψ ( x ) = e − i k ⋅ x ψ ( k ) {\displaystyle \psi (x)=e^{-ik\cdot x}\psi (k)} , applying the on-shell constraint that k ⋅ k = 0 {\displaystyle k\cdot k=0} and normalizing the momentum to be a 3D unit vector: k ^ i = k i / k 0 {\displaystyle {\hat {k}}_{i}=k_{i}/k_{0}} to write Examining

6012-468: A quantum field naturally generalizes to any situation where one can enumerate the continuous symmetries of the system, and define duals in a coherent, consistent fashion. The pairing ties together opposite charges in the fully abstract sense. In physics, a charge is associated with a generator of a continuous symmetry. Different charges are associated with different eigenspaces of the Casimir invariants of

6179-478: A single force at very high energies on a minuscule scale, the Planck scale , but particle accelerators cannot produce the enormous energies required to experimentally probe this. Devising a common theoretical framework that would explain the relation between the forces in a single theory is perhaps the greatest goal of today's theoretical physicists . The weak and electromagnetic forces have already been unified with

6346-467: A structure on a U(1) fiber bundle , the so-called circle bundle . This provides a geometric interpretation of electromagnetism: the electromagnetic potential A μ {\displaystyle A_{\mu }} is interpreted as the gauge connection (the Ehresmann connection ) on the circle bundle. This geometric interpretation then allows (literally almost) anything possessing

6513-442: A theoretical basis for electromagnetic behavior such as quantum tunneling , in which a certain percentage of electrically charged particles move in ways that would be impossible under the classical electromagnetic theory, that is necessary for everyday electronic devices such as transistors to function. The weak interaction or weak nuclear force is responsible for some nuclear phenomena such as beta decay . Electromagnetism and

6680-691: A vector space such that any one of them can be singled out at any given time, via the creation and annihilation operators. The creation and annihilation operators obey the canonical commutation relations , in that the one operator "undoes" what the other "creates". This implies that any given solution u ( p → , σ , n ) {\displaystyle u\left({\vec {p}},\sigma ,n\right)} must be paired with its "anti-solution" v ( p → , σ , n ) {\displaystyle v\left({\vec {p}},\sigma ,n\right)} so that one undoes or cancels out

6847-455: A way that it remains consistently dual when integrating over the fiber of the frame bundle, when integrating (summing) over the fiber that describes the spin, and when integrating (summing) over any other fibers that occur in the theory. When the fiber to be integrated over is the U(1) fiber of electromagnetism, the dual pairing is such that the direction (orientation) on the fiber is reversed. When

C-symmetry - Misplaced Pages Continue

7014-420: Is n + 1 {\displaystyle n+1} points in general linear position . A projective basis is n + 2 {\displaystyle n+2} points in general position, in a projective space of dimension n . A convex basis of a polytope is the set of the vertices of its convex hull . A cone basis consists of one point by edge of a polygonal cone. See also

7181-572: Is a linear isomorphism from the vector space F n {\displaystyle F^{n}} onto V . In other words, F n {\displaystyle F^{n}} is the coordinate space of V , and the n -tuple φ − 1 ( v ) {\displaystyle \varphi ^{-1}(\mathbf {v} )} is the coordinate vector of v . The inverse image by φ {\displaystyle \varphi } of b i {\displaystyle \mathbf {b} _{i}}

7348-443: Is a basis if it satisfies the two following conditions: The scalars a i {\displaystyle a_{i}} are called the coordinates of the vector v with respect to the basis B , and by the first property they are uniquely determined. A vector space that has a finite basis is called finite-dimensional . In this case, the finite subset can be taken as B itself to check for linear independence in

7515-416: Is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B . In other words, a basis is a linearly independent spanning set . A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of

7682-413: Is a basis of V . Since L max belongs to X , we already know that L max is a linearly independent subset of V . If there were some vector w of V that is not in the span of L max , then w would not be an element of L max either. Let L w = L max ∪ { w } . This set is an element of X , that is, it is a linearly independent subset of V (because w

7849-419: Is a bit subtle, and requires articulation. It is often said that charge conjugation does not alter the chirality of particles. This is not the case for fields , the difference arising in the "hole theory" interpretation of particles, where an anti-particle is interpreted as the absence of a particle. This is articulated below. Conventionally, γ 5 {\displaystyle \gamma _{5}}

8016-400: Is a manifestation of the so-called measure concentration phenomenon . The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n -dimensional cube [−1, 1] as a function of dimension, n . A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If

8183-423: Is a pairing of the two. This sketch also provides enough hints to indicate what charge conjugation might look like in a general geometric setting. There is no particular forced requirement to use perturbation theory, to construct quantum fields that will act as middle-men in a perturbative expansion. Charge conjugation can be given a general setting. For general Riemannian and pseudo-Riemannian manifolds , one has

8350-734: Is also called a coordinate frame or simply a frame (for example, a Cartesian frame or an affine frame ). Let, as usual, F n {\displaystyle F^{n}} be the set of the n -tuples of elements of F . This set is an F -vector space, with addition and scalar multiplication defined component-wise. The map φ : ( λ 1 , … , λ n ) ↦ λ 1 b 1 + ⋯ + λ n b n {\displaystyle \varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n}}

8517-426: Is an area of active research. It is hypothesized that gravitation is mediated by a massless spin-2 particle called the graviton . Although general relativity has been experimentally confirmed (at least for weak fields, i.e. not black holes) on all but the smallest scales, there are alternatives to general relativity . These theories must reduce to general relativity in some limit, and the focus of observational work

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8684-521: Is an element of X . Therefore, L Y is an upper bound for Y in ( X , ⊆) : it is an element of X , that contains every element of Y . As X is nonempty, and every totally ordered subset of ( X , ⊆) has an upper bound in X , Zorn's lemma asserts that X has a maximal element. In other words, there exists some element L max of X satisfying the condition that whenever L max ⊆ L for some element L of X , then L = L max . It remains to prove that L max

8851-529: Is any real number. A simple basis of this vector space consists of the two vectors e 1 = (1, 0) and e 2 = (0, 1) . These vectors form a basis (called the standard basis ) because any vector v = ( a , b ) of R may be uniquely written as v = a e 1 + b e 2 . {\displaystyle \mathbf {v} =a\mathbf {e} _{1}+b\mathbf {e} _{2}.} Any other pair of linearly independent vectors of R , such as (1, 1) and (−1, 2) , forms also

9018-641: Is associated with a v ( p → ) {\displaystyle v\left({\vec {p}}\right)} of the opposite momentum and energy. The quantum field is also a sum over all possible spin states; the dual pairing again matching opposite spins. Likewise for any other quantum numbers, these are also paired as opposites. There is a technical difficulty in carrying out this dual pairing: one must describe what it means for some given solution u {\displaystyle u} to be "dual to" some other solution v , {\displaystyle v,} and to describe it in such

9185-404: Is associated with the determinant bundle of the spin structure, effectively tying together the left and right-handed spinors through complex conjugation. What remains is to work through the discrete symmetries of the above construction. There are several that appear to generalize P-symmetry and T-symmetry . Identifying the p {\displaystyle p} dimensions with time, and

9352-457: Is bigger in that it has a double covering by S O ( p , q ) × U ( 1 ) . {\displaystyle SO(p,q)\times U(1).} The U ( 1 ) {\displaystyle U(1)} piece can be identified with electromagnetism in several different ways. One way is that the Dirac operators on the spin manifold, when squared, contain

9519-440: Is constant no matter how fast the observer is moving, showed that the theoretical result implied by Maxwell's equations has profound implications far beyond electromagnetism on the very nature of time and space. In another work that departed from classical electro-magnetism, Einstein also explained the photoelectric effect by utilizing Max Planck's discovery that light was transmitted in 'quanta' of specific energy content based on

9686-469: Is customary to refer to B o l d {\displaystyle B_{\mathrm {old} }} and B n e w {\displaystyle B_{\mathrm {new} }} as the old basis and the new basis , respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has expressions involving the old coordinates, and if one wants to obtain equivalent expressions in terms of

9853-421: Is denoted, as usual, by ⊆ . Let Y be a subset of X that is totally ordered by ⊆ , and let L Y be the union of all the elements of Y (which are themselves certain subsets of V ). Since ( Y , ⊆) is totally ordered, every finite subset of L Y is a subset of an element of Y , which is a linearly independent subset of V , and hence L Y is linearly independent. Thus L Y

10020-489: Is described in the article on C-parity . Charge conjugation occurs as a symmetry in three different but closely related settings: a symmetry of the (classical, non-quantized) solutions of several notable differential equations, including the Klein–Gordon equation and the Dirac equation , a symmetry of the corresponding quantum fields, and in a general setting, a symmetry in (pseudo-) Riemannian geometry . In all three cases,

10187-732: Is equal to 1, is a countable Hamel basis. In the study of Fourier series , one learns that the functions {1} ∪ { sin( nx ), cos( nx ) : n = 1, 2, 3, ... } are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions f satisfying ∫ 0 2 π | f ( x ) | 2 d x < ∞ . {\displaystyle \int _{0}^{2\pi }\left|f(x)\right|^{2}\,dx<\infty .} The functions {1} ∪ { sin( nx ), cos( nx ) : n = 1, 2, 3, ... } are linearly independent, and every function f that

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10354-535: Is exactly one polynomial of each degree (such as the Bernstein basis polynomials or Chebyshev polynomials ) is also a basis. (Such a set of polynomials is called a polynomial sequence .) But there are also many bases for F [ X ] that are not of this form. Many properties of finite bases result from the Steinitz exchange lemma , which states that, for any vector space V , given a finite spanning set S and

10521-506: Is exactly what one wants to make contact with the Pauli exclusion principle . Another is the existence of a chiral element , analogous to the gamma matrix γ 5 {\displaystyle \gamma _{5}} which sorts these spinors into left and right-handed subspaces. The complexification is a key ingredient, and it provides "electromagnetism" in this generalized setting. The spinor bundle doesn't "just" transform under

10688-444: Is given by polynomial rings . If F is a field, the collection F [ X ] of all polynomials in one indeterminate X with coefficients in F is an F -vector space. One basis for this space is the monomial basis B , consisting of all monomials : B = { 1 , X , X 2 , … } . {\displaystyle B=\{1,X,X^{2},\ldots \}.} Any set of polynomials such that there

10855-789: Is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If X is an infinite-dimensional normed vector space that is complete (i.e. X is a Banach space ), then any Hamel basis of X is necessarily uncountable . This is a consequence of the Baire category theorem . The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional ( non-complete ) normed spaces that have countable Hamel bases. Consider c 00 {\displaystyle c_{00}} ,

11022-466: Is left–right asymmetric. The weak interaction even violates CP symmetry but does conserve CPT . The strong interaction , or strong nuclear force , is the most complicated interaction, mainly because of the way it varies with distance. The nuclear force is powerfully attractive between nucleons at distances of about 1 femtometre (fm, or 10 metres), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. At distances less than 0.7 fm,

11189-473: Is meant by the "maximal violation" of C-symmetry in the weak interaction. Some postulated extensions of the Standard Model , like left-right models , restore this C-symmetry. The Dirac field has a "hidden" U ( 1 ) {\displaystyle U(1)} gauge freedom, allowing it to couple directly to the electromagnetic field without any further modifications to the Dirac equation or

11356-678: Is not given an explicit symbolic name, when applied to single-particle solutions of the Dirac equation. This is in contrast to the case when the quantized field is discussed, where a unitary operator C {\displaystyle {\mathcal {C}}} is defined (as done in a later section, below). For the present section, let the involution be named as C : ψ ↦ ψ c {\displaystyle {\mathsf {C}}:\psi \mapsto \psi ^{c}} so that C ψ = ψ c . {\displaystyle {\mathsf {C}}\psi =\psi ^{c}.} Taking this to be

11523-442: Is not in the span of L max , and L max is independent). As L max ⊆ L w , and L max ≠ L w (because L w contains the vector w that is not contained in L max ), this contradicts the maximality of L max . Thus this shows that L max spans V . Hence L max is linearly independent and spans V . It is thus a basis of V , and this proves that every vector space has

11690-890: Is obtained in the Majorana basis. A worked example follows. For the case of massless Dirac spinor fields, chirality is equal to helicity for the positive energy solutions (and minus the helicity for negative energy solutions). One obtains this by writing the massless Dirac equation as Multiplying by γ 5 γ 0 = − i γ 1 γ 2 γ 3 {\displaystyle \gamma ^{5}\gamma ^{0}=-i\gamma ^{1}\gamma ^{2}\gamma ^{3}} one obtains where σ μ ν = i [ γ μ , γ ν ] / 2 {\displaystyle \sigma ^{\mu \nu }=i\left[\gamma ^{\mu },\gamma ^{\nu }\right]/2}

11857-439: Is often useful to express the coordinates of a vector x with respect to B o l d {\displaystyle B_{\mathrm {old} }} in terms of the coordinates with respect to B n e w . {\displaystyle B_{\mathrm {new} }.} This can be done by the change-of-basis formula , that is described below. The subscripts "old" and "new" have been chosen because it

12024-399: Is particularly troublesome, physically, as the universe is primarily filled with matter , not anti-matter , whereas the naive C-symmetry of the physical laws suggests that there should be equal amounts of both. It is currently believed that CP-violation during the early universe can account for the "excess" matter, although the debate is not settled. Earlier textbooks on cosmology , predating

12191-743: Is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that lim n → ∞ ∫ 0 2 π | a 0 + ∑ k = 1 n ( a k cos ⁡ ( k x ) + b k sin ⁡ ( k x ) ) − f ( x ) | 2 d x = 0 {\displaystyle \lim _{n\to \infty }\int _{0}^{2\pi }{\biggl |}a_{0}+\sum _{k=1}^{n}\left(a_{k}\cos \left(kx\right)+b_{k}\sin \left(kx\right)\right)-f(x){\biggr |}^{2}dx=0} for suitable (real or complex) coefficients

12358-513: Is that not every module has a basis. A module that has a basis is called a free module . Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions . A module over the integers is exactly the same thing as an abelian group . Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of

12525-479: Is the angular momentum operator and ϵ i j k {\displaystyle \epsilon _{ijk}} is the totally antisymmetric tensor . This can be brought to a slightly more recognizable form by defining the 3D spin operator Σ m ≡ ϵ i j m σ i j , {\displaystyle \Sigma ^{m}\equiv {\epsilon _{ij}}^{m}\sigma ^{ij},} taking

12692-431: Is the gluon , traversing minuscule distance among quarks, is modeled in quantum chromodynamics (QCD). EWT, QCD, and the Higgs mechanism comprise particle physics ' Standard Model (SM). Predictions are usually made using calculational approximation methods, although such perturbation theory is inadequate to model some experimental observations (for instance bound states and solitons ). Still, physicists widely accept

12859-404: Is the n -tuple e i {\displaystyle \mathbf {e} _{i}} all of whose components are 0, except the i th that is 1. The e i {\displaystyle \mathbf {e} _{i}} form an ordered basis of F n {\displaystyle F^{n}} , which is called its standard basis or canonical basis . The ordered basis B

13026-407: Is the classical theory of electromagnetism, suitable for most technological purposes. The constant speed of light in vacuum (customarily denoted with a lowercase letter c ) can be derived from Maxwell's equations, which are consistent with the theory of special relativity. Albert Einstein 's 1905 theory of special relativity , however, which follows from the observation that the speed of light

13193-481: Is the image by φ {\displaystyle \varphi } of the canonical basis of F n {\displaystyle F^{n}} . It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of F n {\displaystyle F^{n}} , and that every linear isomorphism from F n {\displaystyle F^{n}} onto V may be defined as

13360-463: Is the momentum operator. Taking the Weyl representation of the gamma matrices, one may write a (now taken to be massive) Dirac spinor as The corresponding dual (anti-particle) field is The charge-conjugate spinors are where, as before, η c {\displaystyle \eta _{c}} is a phase factor that can be taken to be η c = 1. {\displaystyle \eta _{c}=1.} Note that

13527-467: Is the momentum, σ {\displaystyle \sigma } is a spin label, n {\displaystyle n} is an auxiliary label for other states in the system. The a {\displaystyle a} and a † {\displaystyle a^{\dagger }} are creation and annihilation operators ( ladder operators ) and u , v {\displaystyle u,v} are solutions to

13694-533: Is the smallest infinite cardinal, the cardinal of the integers. The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for topological vector spaces – a large class of vector spaces including e.g. Hilbert spaces , Banach spaces , or Fréchet spaces . The preference of other types of bases for infinite-dimensional spaces

13861-420: Is to construct a spin structure . Perhaps the most remarkable thing about this is that it is a very recognizable generalization to a ( p , q ) {\displaystyle (p,q)} -dimensional pseudo-Riemannian manifold of the conventional physics concept of spinors living on a (1,3)-dimensional Minkowski spacetime . The construction passes through a complexified Clifford algebra to build

14028-441: Is to establish limits on what deviations from general relativity are possible. Proposed extra dimensions could explain why the gravity force is so weak. Electromagnetism and weak interaction appear to be very different at everyday low energies. They can be modeled using two different theories. However, above unification energy, on the order of 100 GeV , they would merge into a single electroweak force. The electroweak theory

14195-448: Is to realize that, in low dimensions (in low-dimensional spacetime) there are many "accidental" isomorphisms between various Lie groups and other assorted structures. Being able to examine them in a general setting disentangles these relationships, exposing more clearly "where things come from". The laws of electromagnetism (both classical and quantum ) are invariant under the exchange of electrical charges with their negatives. For

14362-704: Is used as the chirality operator. Under charge conjugation, it transforms as and whether or not γ 5 T {\displaystyle \gamma _{5}^{\textsf {T}}} equals γ 5 {\displaystyle \gamma _{5}} depends on the chosen representation for the gamma matrices. In the Dirac and chiral basis, one does have that γ 5 T = γ 5 {\displaystyle \gamma _{5}^{\textsf {T}}=\gamma _{5}} , while γ 5 T = − γ 5 {\displaystyle \gamma _{5}^{\textsf {T}}=-\gamma _{5}}

14529-435: Is vastly stronger. It is the force that binds electrons to atoms, and it holds molecules together . It is responsible for everyday phenomena like light , magnets , electricity , and friction . Electromagnetism fundamentally determines all macroscopic, and many atomic-level, properties of the chemical elements . In a four kilogram (~1 gallon) jug of water, there is of total electron charge. Thus, if we place two such jugs

14696-523: Is very important for modern cosmology , particularly on how the universe evolved. This is because shortly after the Big Bang, when the temperature was still above approximately 10   K , the electromagnetic force and the weak force were still merged as a combined electroweak force. For contributions to the unification of the weak and electromagnetic interaction between elementary particles , Abdus Salam, Sheldon Glashow and Steven Weinberg were awarded

14863-516: Is ε-orthogonal to y if | ⟨ x , y ⟩ | / ( ‖ x ‖ ‖ y ‖ ) < ε {\displaystyle \left|\left\langle x,y\right\rangle \right|/\left(\left\|x\right\|\left\|y\right\|\right)<\varepsilon } (that is, cosine of the angle between x and y is less than ε ). In high dimensions, two independent random vectors are with high probability almost orthogonal, and

15030-426: The q {\displaystyle q} dimensions with space, one can reverse the tangent vectors in the p {\displaystyle p} dimensional subspace to get time reversal, and flipping the direction of the q {\displaystyle q} dimensions corresponds to parity. The C-symmetry can be identified with the reflection on the line bundle. To tie all of these together into

15197-464: The Higgs boson were originally mixed components of a different set of ancient pre-symmetry-breaking fields. As the early universe cooled, these fields split into the long-range electromagnetic interaction, the short-range weak interaction, and the Higgs boson. In the Higgs mechanism , the Higgs field manifests Higgs bosons that interact with some quantum particles in a way that endows those particles with mass. The strong interaction, whose force carrier

15364-473: The Higgs field 's cubic Yukawa coupling produces a weakly attractive fifth interaction. After spontaneous symmetry breaking via the Higgs mechanism , Yukawa terms remain of the form with Yukawa coupling λ i {\displaystyle \lambda _{i}} , particle mass m i {\displaystyle m_{i}} (in eV ), and Higgs vacuum expectation value 246.22 GeV . Hence coupled particles can exchange

15531-515: The Nobel Prize in Physics in 1979. Electromagnetism is the force that acts between electrically charged particles. This phenomenon includes the electrostatic force acting between charged particles at rest, and the combined effect of electric and magnetic forces acting between charged particles moving relative to each other. Electromagnetism has an infinite range, as gravity does, but

15698-535: The Scientific Revolution , Galileo Galilei experimentally determined that this hypothesis was wrong under certain circumstances—neglecting the friction due to air resistance and buoyancy forces if an atmosphere is present (e.g. the case of a dropped air-filled balloon vs a water-filled balloon), all objects accelerate toward the Earth at the same rate. Isaac Newton's law of Universal Gravitation (1687)

15865-440: The Standard Model of particle physics . Within the Standard Model, the strong interaction is carried by a particle called the gluon and is responsible for quarks binding together to form hadrons , such as protons and neutrons . As a residual effect, it creates the nuclear force that binds the latter particles to form atomic nuclei . The weak interaction is carried by particles called W and Z bosons , and also acts on

16032-436: The Weyl equation , but with opposite energy: and Note the freedom one has to equate negative helicity with negative energy, and thus the anti-particle with the particle of opposite helicity. To be clear, the σ {\displaystyle \sigma } here are the Pauli matrices , and p μ = i ∂ μ {\displaystyle p_{\mu }=i\partial _{\mu }}

16199-431: The axiom of choice or a weaker form of it, such as the ultrafilter lemma . If V is a vector space over a field F , then: If V is a vector space of dimension n , then: Let V be a vector space of finite dimension n over a field F , and B = { b 1 , … , b n } {\displaystyle B=\{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}} be

16366-725: The column vectors of the coordinates of v in the old and the new basis respectively, then the formula for changing coordinates is X = A Y . {\displaystyle X=AY.} The formula can be proven by considering the decomposition of the vector x on the two bases: one has x = ∑ i = 1 n x i v i , {\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {v} _{i},} and x = ∑ j = 1 n y j w j = ∑ j = 1 n y j ∑ i = 1 n

16533-513: The coordinates of v over B . However, if one talks of the set of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same set of coefficients. For example, 3 b 1 + 2 b 2 {\displaystyle 3\mathbf {b} _{1}+2\mathbf {b} _{2}} and 2 b 1 + 3 b 2 {\displaystyle 2\mathbf {b} _{1}+3\mathbf {b} _{2}} have

16700-420: The electroweak theory of Sheldon Glashow , Abdus Salam , and Steven Weinberg , for which they received the 1979 Nobel Prize in physics. Some physicists seek to unite the electroweak and strong fields within what is called a Grand Unified Theory (GUT). An even bigger challenge is to find a way to quantize the gravitational field, resulting in a theory of quantum gravity (QG) which would unite gravity in

16867-483: The fundamental interactions or fundamental forces are interactions in nature that appear not to be reducible to more basic interactions. There are four fundamental interactions known to exist: The gravitational and electromagnetic interactions produce long-range forces whose effects can be seen directly in everyday life. The strong and weak interactions produce forces at subatomic scales and govern nuclear interactions inside atoms . Some scientists hypothesize that

17034-452: The n -tuple with all components equal to 0, except the i th, which is 1. Then e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} is a basis of F n , {\displaystyle F^{n},} which is called the standard basis of F n . {\displaystyle F^{n}.} A different flavor of example

17201-498: The pseudo-orthogonal group S O ( p , q ) {\displaystyle SO(p,q)} , the generalization of the Lorentz group S O ( 1 , 3 ) {\displaystyle SO(1,3)} , but under a bigger group, the complexified spin group S p i n C ( p , q ) . {\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).} It

17368-441: The reduced Planck constant ). Since such interactions result in a change in momentum, they can give rise to classical Newtonian forces . In quantum mechanics, physicists often use the terms "force" and "interaction" interchangeably; for example, the weak interaction is sometimes referred to as the "weak force". According to the present understanding, there are four fundamental interactions or forces: gravitation , electromagnetism,

17535-478: The universal enveloping algebra for those symmetries. This is the case for both the Lorentz symmetry of the underlying spacetime manifold , as well as the symmetries of any fibers in the fiber bundle posed above the spacetime manifold. Duality replaces the generator of the symmetry with minus the generator. Charge conjugation is thus associated with reflection along the line bundle or determinant bundle of

17702-417: The weak interaction , and the strong interaction. Their magnitude and behaviour vary greatly, as described in the table below. Modern physics attempts to explain every observed physical phenomenon by these fundamental interactions. Moreover, reducing the number of different interaction types is seen as desirable. Two cases in point are the unification of: Both magnitude ("relative strength") and "range" of

17869-575: The (free, non-interacting, uncoupled) differential equation in question. The quantum field plays a central role because, in general, it is not known how to obtain exact solutions to the system of coupled differential questions. However, via perturbation theory, approximate solutions can be constructed as combinations of the free-field solutions. To perform this construction, one has to be able to extract and work with any one given free-field solution, on-demand, when required. The quantum field provides exactly this: it enumerates all possible free-field solutions in

18036-402: The 1940s to 1960s, and an extremely complicated theory of hadrons as strongly interacting particles was developed. Most notably: While each of these approaches offered insights, no approach led directly to a fundamental theory. Murray Gell-Mann along with George Zweig first proposed fractionally charged quarks in 1961. Throughout the 1960s, different authors considered theories similar to

18203-516: The 1970s, routinely suggested that perhaps distant galaxies were made entirely of anti-matter, thus maintaining a net balance of zero in the universe. This article focuses on exposing and articulating the C-symmetry of various important equations and theoretical systems, including the Dirac equation and the structure of quantum field theory . The various fundamental particles can be classified according to behavior under charge conjugation; this

18370-439: The Standard Model as science's most experimentally confirmed theory. Beyond the Standard Model , some theorists work to unite the electroweak and strong interactions within a Grand Unified Theory (GUT). Some attempts at GUTs hypothesize "shadow" particles, such that every known matter particle associates with an undiscovered force particle , and vice versa, altogether supersymmetry (SUSY). Other theorists seek to quantize

18537-561: The Weyl basis, as above, these eigenstates are and The Majorana spinor is conventionally taken as just the positive eigenstate, namely ψ ( + ) . {\displaystyle \psi ^{(+)}.} The chiral operator γ 5 {\displaystyle \gamma _{5}} exchanges these two, in that This is readily verified by direct substitution. Bear in mind that C {\displaystyle {\mathsf {C}}} does not have

18704-434: The above definition. It is often convenient or even necessary to have an ordering on the basis vectors, for example, when discussing orientation , or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to the basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering

18871-733: The above translates to This directly demonstrates that charge conjugation, applied to single-particle complex-number-valued solutions of the Dirac equation flips the chirality of the solution. The projectors onto the charge conjugation eigenspaces are P ( + ) = ( 1 + C ) P L {\displaystyle P^{(+)}=(1+{\mathsf {C}})P_{\text{L}}} and P ( − ) = ( 1 − C ) P R . {\displaystyle P^{(-)}=(1-{\mathsf {C}})P_{\text{R}}.} The phase factor   η c   {\displaystyle \ \eta _{c}\ } can be given

19038-425: The above, one concludes that angular momentum eigenstates ( helicity eigenstates) correspond to eigenstates of the chiral operator . This allows the massless Dirac field to be cleanly split into a pair of Weyl spinors ψ L {\displaystyle \psi _{\text{L}}} and ψ R , {\displaystyle \psi _{\text{R}},} each individually satisfying

19205-435: The angle between the vectors was within π/2 ± 0.037π/2 then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within π/2 ± 0.037π/2 then the vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of

19372-416: The associated potential, as given in the table, are meaningful only within a rather complex theoretical framework. The table below lists properties of a conceptual scheme that remains the subject of ongoing research. The modern (perturbative) quantum mechanical view of the fundamental forces other than gravity is that particles of matter ( fermions ) do not directly interact with each other, but rather carry

19539-396: The basis elements. In order to emphasize that an order has been chosen, one speaks of an ordered basis , which is therefore not simply an unstructured set , but a sequence , an indexed family , or similar; see § Ordered bases and coordinates below. The set R of the ordered pairs of real numbers is a vector space under the operations of component-wise addition (

19706-627: The case of electrons and quarks , both of which are fundamental particle fermion fields, the single-particle field excitations are described by the Dirac equation One wishes to find a charge-conjugate solution A handful of algebraic manipulations are sufficient to obtain the second from the first. Standard expositions of the Dirac equation demonstrate a conjugate field ψ ¯ = ψ † γ 0 , {\displaystyle {\overline {\psi }}=\psi ^{\dagger }\gamma ^{0},} interpreted as an anti-particle field, satisfying

19873-411: The case of the real numbers R viewed as a vector space over the field Q of rational numbers, Hamel bases are uncountable, and have specifically the cardinality of the continuum, which is the cardinal number 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} , where ℵ 0 {\displaystyle \aleph _{0}} ( aleph-nought )

20040-435: The chain) is recorded. For each n , 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented. Let V be any vector space over some field F . Let X be the set of all linearly independent subsets of V . The set X is nonempty since the empty set is an independent subset of V , and it is partially ordered by inclusion, which

20207-460: The charge conjugation matrix, has an explicit form given in the article on gamma matrices . Curiously, this form is not representation-independent, but depends on the specific matrix representation chosen for the gamma group (the subgroup of the Clifford algebra capturing the algebraic properties of the gamma matrices ). This matrix is representation dependent due to a subtle interplay involving

20374-453: The complex-transposed Dirac equation Note that some but not all of the signs have flipped. Transposing this back again gives almost the desired form, provided that one can find a 4×4 matrix C {\displaystyle C} that transposes the gamma matrices to insert the required sign-change: The charge conjugate solution is then given by the involution The 4×4 matrix C , {\displaystyle C,} called

20541-503: The complexification of the spin group describing the Lorentz covariance of charged particles. The complex number η c {\displaystyle \eta _{c}} is an arbitrary phase factor | η c | = 1 , {\displaystyle |\eta _{c}|=1,} generally taken to be η c = 1. {\displaystyle \eta _{c}=1.} The interplay between chirality and charge conjugation

20708-515: The context of infinite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (named after Georg Hamel ) or algebraic basis can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are orthogonal bases on Hilbert spaces , Schauder bases , and Markushevich bases on normed linear spaces . In

20875-437: The continuous symmetries, the interpretation of the discrete symmetries is a bit more intellectually demanding and confusing. An early surprise appeared in the 1950s, when Chien Shiung Wu demonstrated that the weak interaction violated P-symmetry. For several decades, it appeared that the combined symmetry CP was preserved, until CP-violating interactions were discovered. Both discoveries lead to Nobel prizes . The C-symmetry

21042-400: The coordinates of a vector x over the old and the new basis respectively, the change-of-basis formula is x i = ∑ j = 1 n a i , j y j , {\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},} for i = 1, ..., n . This formula may be concisely written in matrix notation. Let A be

21209-422: The creation or destruction of particles: see Feynman diagrams for examples. Gravitation is the weakest of the four interactions at the atomic scale, where electromagnetic interactions dominate. Gravitation is the most important of the four fundamental forces for astronomical objects over astronomical distances for two reasons. First, gravitation has an infinite effective range, like electromagnetism but unlike

21376-424: The definition of a vector space by a ring , one gets the definition of a module . For modules, linear independence and spanning sets are defined exactly as for vector spaces, although " generating set " is more commonly used than that of "spanning set". Like for vector spaces, a basis of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces

21543-477: The discrete symmetry z = ( x + i y ) ↦ z ¯ = ( x − i y ) {\displaystyle z=(x+iy)\mapsto {\overline {z}}=(x-iy)} that performs complex conjugation, that reverses the sense of direction around the circle. In quantum field theory , charge conjugation can be understood as the exchange of particles with anti-particles . To understand this statement, one must have

21710-488: The electromagnetic field's force carrier, which if unimpeded traverse potentially infinite distance. Electromagnetism's QFT is quantum electrodynamics (QED). The force carriers of the weak interaction are the massive W and Z bosons . Electroweak theory (EWT) covers both electromagnetism and the weak interaction. At the high temperatures shortly after the Big Bang , the weak interaction, the electromagnetic interaction, and

21877-597: The electromagnetic field—then it could be reconciled with Galilean relativity and Newton's laws. (However, such a "Maxwell aether" was later disproven; Newton's laws did, in fact, have to be replaced.) The Standard Model of particle physics was developed throughout the latter half of the 20th century. In the Standard Model, the electromagnetic, strong, and weak interactions associate with elementary particles , whose behaviours are modelled in quantum mechanics (QM). For predictive success with QM's probabilistic outcomes, particle physics conventionally models QM events across

22044-412: The electromagnetic force is far stronger than gravity, it tends to cancel itself out within large objects, so over large (astronomical) distances gravity tends to be the dominant force, and is responsible for holding together the large scale structures in the universe, such as planets, stars, and galaxies. Many theoretical physicists believe these fundamental forces to be related and to become unified into

22211-558: The electrons in jug B, resulting in no net force. Electromagnetic forces are tremendously stronger than gravity, but tend to cancel out so that for astronomical-scale bodies, gravity dominates. Electrical and magnetic phenomena have been observed since ancient times, but it was only in the 19th century James Clerk Maxwell discovered that electricity and magnetism are two aspects of the same fundamental interaction. By 1864, Maxwell's equations had rigorously quantified this unified interaction. Maxwell's theory, restated using vector calculus ,

22378-416: The equation det[ x 1 ⋯ x n ] = 0 (zero determinant of the matrix with columns x i ), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases. It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product , x

22545-412: The fermions, thereby changing their speed and direction. The exchange may also transport a charge between the fermions, changing the charges of the fermions in the process (e.g., turn them from one type of fermion to another). Since bosons carry one unit of angular momentum, the fermion's spin direction will flip from + 1 ⁄ 2 to − 1 ⁄ 2 (or vice versa) during such an exchange (in units of

22712-465: The fiber to be integrated over is the SU(3) fiber of the color charge , the dual pairing again reverses orientation. This "just works" for SU(3) because it has two dual fundamental representations 3 {\displaystyle \mathbf {3} } and 3 ¯ {\displaystyle {\overline {\mathbf {3} }}} which can be naturally paired. This prescription for

22879-512: The field and its charge conjugate, namely that they must be equal: ψ = ψ c . {\displaystyle \psi =\psi ^{c}.} This is perhaps best stated as the requirement that the Majorana spinor must be an eigenstate of the charge conjugation involution. Doing so requires some notational care. In many texts discussing charge conjugation, the involution ψ ↦ ψ c {\displaystyle \psi \mapsto \psi ^{c}}

23046-435: The field itself. This is not the case for scalar fields , which must be explicitly "complexified" to couple to electromagnetism. This is done by "tensoring in" an additional factor of the complex plane C {\displaystyle \mathbb {C} } into the field, or constructing a Cartesian product with U ( 1 ) {\displaystyle U(1)} . Fundamental force In physics ,

23213-454: The first principles of QCD, establishing, to a level of confidence tantamount to certainty, that QCD will confine quarks. Since then, QCD has been the established theory of strong interactions. QCD is a theory of fractionally charged quarks interacting by means of 8 bosonic particles called gluons. The gluons also interact with each other, not just with the quarks, and at long distances the lines of force collimate into strings, loosely modeled by

23380-435: The frequency, which we now call photons . Starting around 1927, Paul Dirac combined quantum mechanics with the relativistic theory of electromagnetism . Further work in the 1940s, by Richard Feynman , Freeman Dyson , Julian Schwinger , and Sin-Itiro Tomonaga , completed this theory, which is now called quantum electrodynamics , the revised theory of electromagnetism. Quantum electrodynamics and quantum mechanics provide

23547-579: The gravitational field by the modelling behaviour of its hypothetical force carrier, the graviton and achieve quantum gravity (QG). One approach to QG is loop quantum gravity (LQG). Still other theorists seek both QG and GUT within one framework, reducing all four fundamental interactions to a Theory of Everything (ToE). The most prevalent aim at a ToE is string theory , although to model matter particles , it added SUSY to force particles —and so, strictly speaking, became superstring theory . Multiple, seemingly disparate superstring theories were unified on

23714-826: The isomorphism that maps the canonical basis of F n {\displaystyle F^{n}} onto a given ordered basis of V . In other words, it is equivalent to define an ordered basis of V , or a linear isomorphism from F n {\displaystyle F^{n}} onto V . Let V be a vector space of dimension n over a field F . Given two (ordered) bases B old = ( v 1 , … , v n ) {\displaystyle B_{\text{old}}=(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})} and B new = ( w 1 , … , w n ) {\displaystyle B_{\text{new}}=(\mathbf {w} _{1},\ldots ,\mathbf {w} _{n})} of V , it

23881-401: The left and right states are inter-changed. This can be restored with a parity transformation. Under parity , the Dirac spinor transforms as Under combined charge and parity, one then has Conventionally, one takes η c = 1 {\displaystyle \eta _{c}=1} globally. See however, the note below. The Majorana condition imposes a constraint between

24048-495: The matrix of the a i , j {\displaystyle a_{i,j}} , and X = [ x 1 ⋮ x n ] and Y = [ y 1 ⋮ y n ] {\displaystyle X={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}\quad {\text{and}}\quad Y={\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}} be

24215-504: The modern fundamental theory of quantum chromodynamics (QCD) as simple models for the interactions of quarks. The first to hypothesize the gluons of QCD were Moo-Young Han and Yoichiro Nambu , who introduced the quark color charge. Han and Nambu hypothesized that it might be associated with a force-carrying field. At that time, however, it was difficult to see how such a model could permanently confine quarks. Han and Nambu also assigned each quark color an integer electrical charge, so that

24382-713: The new coordinates; this is obtained by replacing the old coordinates by their expressions in terms of the new coordinates. Typically, the new basis vectors are given by their coordinates over the old basis, that is, w j = ∑ i = 1 n a i , j v i . {\displaystyle \mathbf {w} _{j}=\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}.} If ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} and ( y 1 , … , y n ) {\displaystyle (y_{1},\ldots ,y_{n})} are

24549-416: The nuclear force becomes repulsive. This repulsive component is responsible for the physical size of nuclei, since the nucleons can come no closer than the force allows. After the nucleus was discovered in 1908, it was clear that a new force, today known as the nuclear force, was needed to overcome the electrostatic repulsion , a manifestation of electromagnetism, of the positively charged protons. Otherwise,

24716-461: The nucleus could not exist. Moreover, the force had to be strong enough to squeeze the protons into a volume whose diameter is about 10 m , much smaller than that of the entire atom. From the short range of this force, Hideki Yukawa predicted that it was associated with a massive force particle, whose mass is approximately 100 MeV. The 1947 discovery of the pion ushered in the modern era of particle physics. Hundreds of hadrons were discovered from

24883-406: The nucleus of atoms , mediating radioactive decay . The electromagnetic force, carried by the photon , creates electric and magnetic fields , which are responsible for the attraction between orbital electrons and atomic nuclei which holds atoms together, as well as chemical bonding and electromagnetic waves , including visible light , and forms the basis for electrical technology. Although

25050-621: The number of independent random vectors, which all are with given high probability pairwise almost orthogonal, grows exponentially with dimension. More precisely, consider equidistribution in n -dimensional ball. Choose N independent random vectors from a ball (they are independent and identically distributed ). Let θ be a small positive number. Then for N random vectors are all pairwise ε-orthogonal with probability 1 − θ . This N growth exponentially with dimension n and N ≫ n {\displaystyle N\gg n} for sufficiently big n . This property of random bases

25217-471: The other. The pairing is to be performed so that all symmetries are preserved. As one is generally interested in Lorentz invariance , the quantum field contains an integral over all possible Lorentz coordinate frames, written above as an integral over all possible momenta (it is an integral over the fiber of the frame bundle ). The pairing requires that a given u ( p → ) {\displaystyle u\left({\vec {p}}\right)}

25384-401: The particle fields, expressed as where the non-calligraphic   C   {\displaystyle \ C\ } is the same 4×4 matrix given before. Charge conjugation does not alter the chirality of particles. A left-handed neutrino would be taken by charge conjugation into a left-handed antineutrino , which does not interact in the Standard Model. This property is what

25551-464: The phase would be constrained to not vary under boosts and rotations. The above describes charge conjugation for the single-particle solutions only. When the Dirac field is second-quantized , as in quantum field theory , the spinor and electromagnetic fields are described by operators. The charge conjugation involution then manifests as a unitary operator C {\displaystyle {\mathcal {C}}} (in calligraphic font) acting on

25718-398: The principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frames of reference . A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C ) is a linearly independent subset of V that spans V . This means that a subset B of V

25885-402: The property of asymptotic freedom , allowing them to make contact with experimental evidence . They concluded that QCD was the complete theory of the strong interactions, correct at all distance scales. The discovery of asymptotic freedom led most physicists to accept QCD since it became clear that even the long-distance properties of the strong interactions could be consistent with experiment if

26052-438: The quarks are permanently confined : the strong force increases indefinitely with distance, trapping quarks inside the hadrons. Assuming that quarks are confined, Mikhail Shifman , Arkady Vainshtein and Valentine Zakharov were able to compute the properties of many low-lying hadrons directly from QCD, with only a few extra parameters to describe the vacuum. In 1980, Kenneth G. Wilson published computer calculations based on

26219-475: The quarks were fractionally charged only on average, and they did not expect the quarks in their model to be permanently confined. In 1971, Murray Gell-Mann and Harald Fritzsch proposed that the Han/Nambu color gauge field was the correct theory of the short-distance interactions of fractionally charged quarks. A little later, David Gross , Frank Wilczek , and David Politzer discovered that this theory had

26386-425: The same set of coefficients {2, 3} , and are different. It is therefore often convenient to work with an ordered basis ; this is typically done by indexing the basis elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis, especially when used in conjunction with an origin ,

26553-567: The space of symmetries. The above then is a sketch of the general idea of a quantum field in quantum field theory. The physical interpretation is that solutions u ( p → , σ , n ) {\displaystyle u\left({\vec {p}},\sigma ,n\right)} correspond to particles, and solutions v ( p → , σ , n ) {\displaystyle v\left({\vec {p}},\sigma ,n\right)} correspond to antiparticles, and so charge conjugation

26720-436: The space of the sequences x = ( x n ) {\displaystyle x=(x_{n})} of real numbers that have only finitely many non-zero elements, with the norm ‖ x ‖ = sup n | x n | {\textstyle \|x\|=\sup _{n}|x_{n}|} . Its standard basis , consisting of the sequences having only one non-zero element, which

26887-543: The strong and weak interactions. Second, gravity always attracts and never repels; in contrast, astronomical bodies tend toward a near-neutral net electric charge, such that the attraction to one type of charge and the repulsion from the opposite charge mostly cancel each other out. Even though electromagnetism is far stronger than gravitation, electrostatic attraction is not relevant for large celestial bodies, such as planets, stars, and galaxies, simply because such bodies contain equal numbers of protons and electrons and so have

27054-533: The structure of galaxies and black holes and, being only attractive, it retards the expansion of the universe . Gravitation also explains astronomical phenomena on more modest scales, such as planetary orbits , as well as everyday experience: objects fall; heavy objects act as if they were glued to the ground, and animals can only jump so high. Gravitation was the first interaction to be described mathematically. In ancient times, Aristotle hypothesized that objects of different masses fall at different rates. During

27221-543: The symmetry is ultimately revealed to be a symmetry under complex conjugation , although exactly what is being conjugated where can be at times obfuscated, depending on notation, coordinate choices and other factors. The charge conjugation symmetry is interpreted as that of electrical charge , because in all three cases (classical, quantum and geometry), one can construct Noether currents that resemble those of classical electrodynamics . This arises because electrodynamics itself, via Maxwell's equations , can be interpreted as

27388-408: The uniqueness of the decomposition of a vector over a basis, here B old {\displaystyle B_{\text{old}}} ; that is x i = ∑ j = 1 n a i , j y j , {\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},} for i = 1, ..., n . If one replaces the field occurring in

27555-407: The weak force are now understood to be two aspects of a unified electroweak interaction — this discovery was the first step toward the unified theory known as the Standard Model . In the theory of the electroweak interaction, the carriers of the weak force are the massive gauge bosons called the W and Z bosons . The weak interaction is the only known interaction that does not conserve parity ; it

27722-427: Was a good approximation of the behaviour of gravitation. Present-day understanding of gravitation stems from Einstein's General Theory of Relativity of 1915, a more accurate (especially for cosmological masses and distances) description of gravitation in terms of the geometry of spacetime . Merging general relativity and quantum mechanics (or quantum field theory ) into a more general theory of quantum gravity

27889-422: Was light, travelling at constant speed in vacuum. If his electromagnetic field theory held true in all inertial frames of reference , this would contradict Newton's theory of motion, which relied on Galilean relativity . If, instead, his field theory only applied to reference frames at rest relative to a mechanical luminiferous aether —presumed to fill all space whether within matter or in vacuum and to manifest

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