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114-490: 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

228-406: 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

342-506: 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

456-661: 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

570-428: A , b ) = ( λ a , λ b ) , {\displaystyle \lambda (a,b)=(\lambda a,\lambda b),} where λ {\displaystyle \lambda } 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 = (

684-564: 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

798-438: 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

912-437: 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 ( a , b ) + ( c , d ) = ( a + c , b + d ) {\displaystyle (a,b)+(c,d)=(a+c,b+d)} and scalar multiplication λ (

1026-457: 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

1140-410: 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 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

1254-477: 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 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

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1368-800: 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

1482-463: 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 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

1596-420: 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

1710-410: 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

1824-538: 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

1938-451: 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 ,

2052-388: 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

2166-413: 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

2280-409: 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

2394-479: 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

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2508-484: 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}}}

2622-472: 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

2736-559: 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

2850-463: 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

2964-467: 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 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

3078-462: 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

3192-417: 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 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,

3306-691: 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

3420-452: 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

3534-420: 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

C-symmetry - Misplaced Pages Continue

3648-572: 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}}

3762-424: 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 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

3876-413: 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

3990-419: 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}}

4104-400: 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

4218-423: 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

4332-505: 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 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}}

4446-651: 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 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

4560-521: 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

4674-641: 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

C-symmetry - Misplaced Pages Continue

4788-404: 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

4902-453: 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

5016-453: 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 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

5130-469: 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

5244-421: 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

5358-489: 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,

5472-732: 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

5586-502: 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

5700-777: 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}} ,

5814-466: 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

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5928-678: 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

6042-442: 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

6156-885: 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}

6270-439: 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

6384-399: 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

6498-743: 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

6612-513: 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

6726-479: 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

6840-450: 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 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

6954-404: 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

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7068-481: 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

7182-463: 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

7296-467: 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

7410-533: 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

7524-420: 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

7638-448: 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

7752-494: 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 , 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

7866-704: 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}}

7980-516: 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

8094-426: 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

8208-436: 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 }}

8322-725: 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

8436-493: 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

8550-473: 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

8664-575: 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

8778-510: 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

8892-557: 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

9006-491: The Wikimedia System Administrators, please include the details below. Request from 172.68.168.226 via cp1108 cp1108, Varnish XID 762741742 Upstream caches: cp1108 int Error: 429, Too Many Requests at Fri, 29 Nov 2024 05:32:10 GMT Coordinate frame In mathematics , 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

9120-733: 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

9234-425: 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

9348-435: 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

9462-419: 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 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

9576-627: 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

9690-411: 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 )

9804-435: 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

9918-455: 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

10032-413: The coefficients λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are scalars (that is, elements of F ), which are called 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

10146-453: 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

10260-503: 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

10374-515: 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

10488-437: 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

10602-400: 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

10716-424: 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

10830-477: 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

10944-416: 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

11058-461: 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

11172-512: 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}}

11286-468: 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 Too Many Requests If you report this error to

11400-826: 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

11514-401: 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

11628-495: 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

11742-713: 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

11856-621: 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

11970-471: 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)}

12084-458: 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

12198-464: 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

12312-450: 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 the same set of coefficients {2, 3} , and are different. It is therefore often convenient to work with an ordered basis ; this

12426-423: 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 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

12540-536: 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}}

12654-567: 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

12768-436: 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

12882-543: 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

12996-408: 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

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