How Cliord algebra can help understand second quantization of fermion and boson elds N.S. Manko c Bor stnik1

2025-04-27 0 0 767.4KB 43 页 10玖币
侵权投诉
How Clifford algebra can help understand second quantization
of fermion and boson fields
N.S. Mankoˇc Borˇstnik1
1Department of Physics, University of Ljubljana
SI-1000 Ljubljana, Slovenia
October 13, 2022
Abstract
In the review article in Progress in Particle and Nuclear Physics [4] the authors present a
rather detailed review of the achievements so far of the spin-charge-family theory that offers the
explanation for the observed properties of elementary fermion and boson fields, if the space-time is
higher than d= (3 + 1), it must be d≥(13 + 1), while fermions only interact with gravity. Ref. [4]
presents also an explanation for the second quantization postulates for the fermion fields, since
the internal space of fermions is in this theory described with the ”basis vectors” determined by
the Clifford odd objects. The anticommutativity of the ”basis vectors” namely transfers to their
creation and annihilation operators. This paper shows that the ”basis vectors” determined by the
Clifford even objects, if used to describe the internal space of boson fields, not only manifest all
the known properties of the observed boson fields, but offer as well the explanation for the second
quantization postulates for boson fields. Properties of fermion and boson fields with the internal
spaces described by the Clifford odd and even objects, respectively, are demonstrated on the toy
model with d= (5 + 1).
1 Introduction
In a long series of works [1, 2, 28, 16, 18, 22, 24, 21] the author has found, together with the collabora-
tors ([14, 15, 20, 25, 8, 5, 4] and the references therein), and in long discussions with participants during
the annual workshops ”What comes beyond the standard models”, the phenomenological success with
the model named the spin-charge-family theory with the properties:
a. The internal space of fermions are described by the ”basis vectors” which are superposition of odd
products of anticommuting objects γa’s in d= (13+1) [12, 8, 5, 4]. Correspondingly the ”basis vectors”
of one Lorentz irreducible representation in internal space of fermions, together with their Hermitian
conjugated partners, anticommute, fulfilling (on the vacuum state) all the requirements for the second
quantized fermion fields [12, 8, 14, 15, 4].
b. The second kind of anticommuting objects, ˜γa’s, equip each irreducible representation of odd ”basis
vectors” with the family quantum number [5, 15].
c. Creation operators for single fermion states — which are tensor products, ∗T, of a finite number of
odd ”basis vectors” appearing in 2d
2−1families, each family with 2d
2−1members, and the (continuously)
infinite momentum/coordinate basis applying on the vacuum state [5, 4] — inherit anticommutativity
1
arXiv:2210.06256v1 [physics.gen-ph] 7 Oct 2022
of ”basis vectors”. Creation operators and their Hermitian conjugated partners correspondingly anti-
commute.
d. The Hilbert space of second quantized fermions is represented by the tensor products of all possible
number of creation operators, from zero to infinity [8], applying on a vacuum state.
e. In a simple starting action massless fermions carry only spins and interact with only gravity —
with the vielbeins and the two kinds of the spin connection fields (the gauge fields of momenta, of
Sab =i
4(γaγb−γbγa) and of ˜
Sab =1
4(˜γa˜γb−˜γb˜γa), respectively 1). The starting action includes only
even products of γa’s and ˜
γa’s ([12] and references therein).
f. Spins from higher dimensions, d > (3 + 1), described by γa’s, manifest in d= (3 + 1) all the charges
of the standard model quarks and leptons and antiquarks and antileptons of particular handedness.
g. Gravity — the gauge fields of Sab, ((a, b) = (5,6, ...., d)), with the space index m= (0,1,2,3) —
manifest as the standard model vector gauge fields [20]. The scalar gauge fields of ˜
Sab and of some
of superposition of Sab, with the space index s= (7,8) manifest as the scalar higgs and Yukawa cou-
plings [21, 22, 16, 4], determining mass matrices (of particular symmetry) and correspondingly the
masses of quarks and leptons and of the weak boson fields after (some of) the scalar fields with the
space index (7,8) gain constant values. The scalar gauge fields of ˜
Sab and of Sab with the space index
s= (9,10, ..., 14) and (a, b) = (5,6, ...., d) offer the explanation for the observed matter/antimatter
asymmetry [18, 24, 25, 4, 19] in the universe.
h. The theory predicts the fourth family to the observed three [30, 31, 32, 34, 35] and the stable
fifth family of heavy quarks and leptons. The stable fifth family nucleons offer the explanation for the
appearance of the dark matter. Due to heavy masses of the fifth family quarks the nuclear interaction
among hadrons of the fifth family members is verry different than the ones so far observed [33, 36].
i. The theory offers a new understanding of the second quantized fermion fields (explained in Ref. [4])
as well as of the second quantized boson fields. The second quantization of boson fields, the gauge fields
of the second quantized fermion fields, is the main topic of this paper [6].
j. The theory seems promising to offer a new insight into Feynman diagrams.
The more work is put into the theory the more phenomena the theory is able to explain.
In this paper we shortly overview the description of the internal space of the second quantized
massless fermion fields with the ”basis vectors” which are the superposition of odd products of the
Clifford algebra objects (operators) γa’s. Tensor products of ”basis vectors” with the basis in ordinary
space form the creation operators for fermions which fulfil the anticommutation relations of the Dirac
second quantized fermion fields, without postulating them [9, 10, 11]. We kindly ask the reader to read
the explanations in Ref. [4], Sect. 3.
The ”basis vectors”, which are the superposition of even products of the Clifford algebra objects
γa’s and are in a tensor product , ∗T, with the basis in ordinary space, have all the properties of the
second quantized boson fields, the gauge fields of the corresponding second quantized fermion fields.
The main part of this paper discusses properties of the internal space of the second quantized boson
fields described by the Clifford even ”basis vectors” in interaction among the boson fields themselves
and with the second quantized fermion fields with the internal space described by the Clifford odd
”basis vectors”, Ref. [4].
In both cases when describing the second quantized either fermion or boson fields the creation
operators are considered to be a tensor, ∗T, product of 2d
2−1×2d
2−1of either anticommuting Clifford
odd (in the case of fermion fields) or commuting Clifford even (in the case of boson fields) ”basis vectors”
and of (continuously) infinite commuting basis of ordinary space.
While in the case of the Clifford odd ”basis vectors” the Hermitian conjugated partners belong to
another group with 2d
2−1×2d
2−1members (which is not reachable by either Sab or ˜
Sab) or both, in the
1If there are no fermions present the two kinds of the spin connection fields are uniquely expressible by the vielbeins [12].
2
case of the Clifford even ”basis vectors” each of the two groups with 2d
2−1×2d
2−1members have their
Hermitian conjugated partners among themselves, that is within the group reachable by either Sab or
˜
Sab.
Subects.2.1, 2.2, 2.3 of Sect.2 are a short overview of the Clifford odd and the Clifford even algebra,
used to described in the spin-charge-family theory the internal space of fermions — as already presented
in Ref. ([4] in Sect. 3) — and in this paper in particular the internal space of bosons, as the author started
in Ref. [6]. In Subsect. 2.1 the anticommuting Grassmann algebra and the two Clifford subalgebras,
each algebra with 2 ×2delements, are presented, and the relations among them discussed.
In Subsect. 2.2 the ”basis vectors” of either odd or even character are defined as eigenvectors of all
the members of the Cartan subalgebra of the Lorentz algebra for the Grassmann and the two Clifford
subalgebras ([4], Sect. 3).
The ”basis vectors” are products of nilpotents and projectors, each nilpotent and each projector
is chosen to be the eigenvector of one member of the Cartan subalgebra. The anticommuting ”basis
vectors” have an odd number of nilpotents, the commuting ”basis vectors” have an even number of
nilpotents.
The anticommutation relations (for fermions, with the odd number of nilpotents) and commutation
relations (for bosons, with the even number of nilpotents) are presented.
There are obviously only one kind of fermion fields and correspondingly also of their gauge fields
observed. There is correspondingly no need for two Clifford subalgebras.
In Subsect. 2.3 this problem is solved by the reduction of the two Clifford subalgebras to only
one, what enables also to give the family quantum numbers to the Clifford odd anticommuting ”basis
vectors”, belonging to different irreducible representations of the Lorentz algebra. The reduction en-
ables as well to define to the Clifford even commuting ”basis vectors” the generators of the Lorentz
transformations in the internal space of bosons.
In Subsect. 2.4 the ”basis vectors” for fermions, 2.4.1, and bosons, 2.4.2, are discussed in details for
the ”toy model” in d= (5 + 1) to make differences in the properties of the Clifford odd and Clifford
even ”basis vectors” transparent and correspondingly easier to understand. The algebraic application
of the Clifford even ”basis vectors” on the Clifford odd ”basis vectors” is demonstrated, as well as the
algebraic application of the Clifford even ”basis vectors” on themselves. This subsection is the main
part of the article.
In Subsect. 2.5 the generalization of the Clifford odd and Clifford even ”basis vectors” to any even
dis discussed.
In Sect. 3 the creation operators of the second quantized fermion and boson fields offered by the
spin-charge-family theory are studied.
Sect. 4 reviews shortly what one can learn in this article and what remains to study.
In App. A ”basis vectors” of one family of quarks and leptons are presented.
In App. B some useful relations are presented.
2 Properties of creation and annihilation operators for fermions
and bosons
Second quantization postulates for fermion and boson fields [9, 10, 11] require that the creation and
annihilation operators for fermions and bosons depend on finite number of spins and other quantum
numbers determining internal space of fermions and bosons and on infinite number of momenta (or
coordinates). While fermions carry half integer spins and charges in fundamental representations of the
corresponding groups bosons carry integer spins and charges in adjoint representations of the groups.
Ref. [4] reports in Subsect. 3.3.1. second quantization postulates for fermions.
3
The first quantized fermion states are in the Dirac’s theory vectors which do not anticommute. There
are the creation operators of the second quantized fermion fields which are postulated to anticommute.
The second quantized fermion fields commute with γamatrices, allowing the second quantization of the
Dirac equation which includes the mass term.
Creation and annihilation of boson fields are postulated to fulfil commutation relations.
In the spin-charge-family theory the internal space of fermions and bosons in even dimensional
spaces d= 2(2n+ 1) is described by the algebraic, ∗A, products of d
2nilpotents and projectors, which
are superposition of odd (nilpotents) and even (projectors) numbers of anticommuting operators γa’s.
Nilpotents and projectors are chosen to be eigenvectors of d
2Cartan subalgebra members of the Lorentz
algebra of Sab =i
4{γa, γb}−, determining the internal space of fermions and bosons.
There are two groups of 2d
2−1members appearing in 2d
2−1irreducible representations which have an
odd number of nilpotents (at least one nilpotent and the rest projectors). The members of one of the
groups are (chosen to be) called ”basis vectors”. The other group contains the Hermitian conjugated
partners of the ”basis vectors”. The group of these odd ”basis vectors” have all the properties needed
to describe internal space of fermions.
There are two groups with an even number of nilpotents, each with 2d
2−12d
2−1members. Each of these
two groups have their Hermitian conjugated partners within the same group. Each of the two groups
with an even number of nilpotents have all the properties needed to describe the internal space of boson
fields, as we shall see in this article.
N.S.M.B. made a choice of d= (13 + 1) since for such dthe theory offers the explanation for all the
assumptions of the standard model, that is for the charges, handedness, families of quarks and leptons
and antiquarks and antileptons, for all the observed vector gauge fields, as well as for the scalar higgs
and Yukawa couplings.
A simple starting action ([4] and the references therein) for the second quantized massless fermion
and antifermion fields, and the corresponding massless boson fields in d= 2(2n+ 1)-dimensional space
is assumed to be
A=Zddx E 1
2(¯
ψ γap0aψ) + h.c. +
Zddx E (α R + ˜α˜
R),
p0a=fαap0α+1
2E{pα, Ef αa}−,
p0α=pα−1
2Sabωabα −1
2˜
Sab ˜ωabα ,
R=1
2{fα[afβb](ωabα,β −ωcaα ωcbβ )}+h.c. ,
˜
R=1
2{fα[afβb](˜ωabα,β −˜ωcaα ˜ωcbβ )}+h.c. . (1)
Here 2fα[afβb]=fαafβb −fαbfβa. The γaoperators appear in the Lagrangean for second quantized
massless fermion fields in pairs.
Fermions, appearing in families, carry only spins and only interact with gravity, what manifests in
d= (3 + 1) as spins and all the observed charges. Vielbeins, fa
α(the gauge field of momenta), and
2fαaare inverted vielbeins to eaαwith the properties eaαfα
b=δa
b, ea
αfβa=δβ
α,E= det(ea
α). Latin indices
a, b, .., m, n, .., s, t, .. denote a tangent space (a flat index), while Greek indices α, β, .., µ, ν, ..σ, τ, .. denote an Einstein index
(a curved index). Letters from the beginning of both the alphabets indicate a general index (a, b, c, .. and α, β, γ, .. ), from
the middle of both the alphabets the observed dimensions 0,1,2,3 (m, n, .. and µ, ν, ..), indexes from the bottom of the al-
phabets indicate the compactified dimensions (s, t, .. and σ, τ, ..). We assume the signature ηab =diag{1,−1,−1,··· ,−1}.
4
two kinds of the spin connection fields, ωabα (the gauge fields of Sab) and ˜ωabα (the gauge fields of ˜
Sab),
manifest in d= (3 + 1) as the known vector gauge fields [20] and the scalar gauge fields taking care of
masses of quarks and leptons and antiquarks and antileptons [4] and the weak boson fields 3.
Internal space of fermions is described in any even dimensional space by ”basis vectors” with an
odd number of nilpotents, the rest are projectors, as mentioned above. Nilpotents, described by an odd
number of γa’s, anticommute with γa, projectors, described by an even number of γa’s, commute with
γa. Correspondingly the ”basis vectors”, describing the internal space of fermions, anticommute among
themselves.
Internal space of bosons is described in any even dimensional space by ”basis vectors” with an even
number of nilpotents and of projectors. The ”basis vectors” describing bosons therefore commute.
Creation operators for either second quantized fermions or bosons are tensor products, ∗T, of the
Clifford odd 2d
2−1×2d
2−1”basis vectors” describing the internal space of fermions or of the Clifford even
2d
2−1×2d
2−1”basis vectors” describing the internal space of bosons, and of an (continuously) infinite
number of basis in ordinary momentum (or coordinate) space.
Creation operators for fermions, represented by anticommuting ”basis vectors” in a tensor product,
∗T, with the basis in ordinary space, anticommute among themselves and with γa’s, creation operators
for bosons, represented by commuting ”basis vectors” in a tensor product, ∗T, with the basis in ordinary
space, commute among themselves and with γa’s.
”Basis vectors” describing the internal space of fermion and boson fields transfer their anticommu-
tativity or commutativity into creation operators, since the basis in ordinary space commute.
One irreducible representation of ”basis vectors”, reachable with Sab, and determining the internal
space of fermions in d= (13 + 1)-dimensional space, includes quarks and leptons and antiquarks and
antileptons, together with the right handed neutrinos and the left handed antineutrinos, as can be
seen in Table 4. There are 64 (= 2d
2−1) members of one irreducible representation, represented as the
eigenvectors of the Cartan subalgebra of the SO(13 + 1) group, analysed with respect to the subgroups
SO(3,1), SU(2), SU(2), SU (3) and U(1), with 7 commuting operators 4.
These Clifford odd anticommuting ”basis vectors” transfer the anti-commutativity to the creation
operators for quarks and leptons and antiquarks and antileptons.
The standard model subgroups (SO(3,1), SU(2), SU (3), U(1)) have one SU(2) group less, and cor-
respondingly the right handed neutrinos and the left handed antinutrinos, having no charge, are in the
standard model assumed not to exist.
Sab’s transform each member of one irreducible representation of fermions into all the members of
the same irreducible representation, ˜
Sab’s transform each member of one irreducible representation to
the same member of another irreducible representation. The postulate, presented in Eq. (16), equips
each irreducible representation with the family quantum number.
The mass terms appear in the spin-charge-family theory after the scalar fields with the space index
s≥5 (s= (7,8) indeed), which are the gauge fields of the two kinds of the spin connection fields ωabs
and ˜ωabs (the gauge fields of Sab and ˜
Sab, respectively), gain the constant values, what makes particu-
lar charges (hypercharge Y=τ4+τ23 and the weak charge τ13, explained in Table 4) non conserved
quantities. The appearance of the mass term in d= (3 + 1) is discussed in Subsects. 6.1, 6.2.2, 7.3 and
7.4 in Ref. [4].
3Since the multiplication with either γa’s or ˜γa’s changes the Clifford odd ”basis vectors” into the Clifford even ”basis
vectors”, and even ”basis vectors” commute, the action for fermions can not include an odd numbers of γa’s or ˜γa’s, what
the simple starting action of Eq. (1) does not. In the starting action γa’s and ˜γa’s appear as γ0γaˆpaor as γ0γcSabωabc
and as γ0γc˜
Sab ˜ωabc.
4The SO(7,1) part is identical for quarks of any of the three colours and for the colourless leptons, and identical for
antiquarks and the colourless antileptons. They differ only in the SO(6) part of the group SO(13,1).
5
摘要：

HowCli ordalgebracanhelpunderstandsecondquantizationoffermionandboson eldsN.S.MankocBorstnik11DepartmentofPhysics,UniversityofLjubljanaSI-1000Ljubljana,SloveniaOctober13,2022AbstractInthereviewarticleinProgressinParticleandNuclearPhysics[4]theauthorspresentaratherdetailedreviewoftheachievementssof...

展开>> 收起<<
How Cliord algebra can help understand second quantization of fermion and boson elds N.S. Manko c Bor stnik1.pdf

共43页,预览5页

还剩页未读， 继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

相关推荐

更多
立即下载
分类：图书资源 价格：10玖币 属性：43 页 大小：767.4KB 格式：PDF 时间：2025-04-27

开通VIP享超值会员特权

  • 多端同步记录
  • 高速下载文档
  • 免费文档工具
  • 分享文档赚钱
  • 每日登录抽奖
  • 优质衍生服务

作者详情

相关内容

更多

热门标签

人际关系 动力学连接体 力的合成 配电装置 高考理综 全宋诗作者索引 公务员考试
/ 43
评分 收藏
分享
立即下载
客服
关注