Download Extension of Lorentz Group Representations for Chiral Fermions

Extension of Lorentz Group Representations for Chiral Fermions
Robert Y. Levine∗
Spectral Sciences, Inc., Burlington, MA, USA
James Y-K. Cho†
School of Mathematical Sciences, Queen Mary, University of London,
Mile End Road, London E1 4NS, UK
(Dated: 18 March 2008)
Abstract
We derive a formulation of the Naimark extension for Dirac spinors. Non-commuting rotation
(spin) and boost generators are extended by first grouping operators into left and right handed
pairs and then defining ancillary spin-1/2 vacuum meters for the three space dimensions. The result
is an explicit example of a recently-proposed theory, in which the extension of the Lorentz group to
commuting operators is an underlying structure for three generations of elementary fermions. We
suggest that the extension appears at the scale of the weak vector bosons, through the CabibboKobayashi-Maskawa matrix in the W ± couplings and SU (2)L ×U (1)-generated left-right mixing in
the Z 0 coupling.
PACS numbers: 03.65.BZ
Keywords: entangled simultaneous quantum measurement, Naimark extension, lepton/quark generations
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Gauge Institute Journal Volume 6, No 1, February 2010
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INTRODUCTION
The principles of quantum measurement are at the foundation of particle physics. For
example, particle spin and momentum assignments are determined by quantum representations of the Lorentz group [1], and quantum electrodynamics as a local U (1) gauge theory
emerges naturally from the phase invariance of quantum observables. However, this close
association of particles and quantum measurement appears unconnected in the broader
standard model. In particular, the existence of three fermionic generations and non-Abelian
gauge theory interactions are independent of quantum representation theory. The specific
generational and isospin structures in the standard model are seemingly not constrained by
quantum theoretical foundations. In this paper we describe the particle states and decay
patterns that arise at the scale of the weak interaction when explicit connections are made
between the standard model and the theories of quantum representation and measurement.
In particular, we show that quantum representation theory can be generalized to explain
fermionic generations from the requirement that the Lorentz group operators are represented on distinguishable fermionic states. Quark and lepton fermionic generations create
representations of the Lorentz group that extend standard particle quantum number assignments. Entanglements of particle quantum states are integral to this extended representation
and involve the weak interaction bosons {Z 0 , W ± }. The Z 0 boson entangles left and right
handed fermions within each generation, and the W ± bosons provide entanglements between generations. The Z 0 entanglement is directly through the SU (2)L ×U (1) fermionic
current, whereas the W ± entanglement mechanism is more subtle. The latter is through the
Cabibbo-Kobayashi-Maskawa (CKM) matrix: right handed quark spectators, in a density
matrix representation of left handed W ± -mediated decays, lead naturally to CKM quark
mixings.
NAIMARK EXTENSIONS
Extended quantum representations and entanglements require augmenting observable operators through the introduction of vacuum ‘meter’ states, in a procedure known as Naimark
extension [2–4]. The basic idea of the extension, applied to a coupled harmonic oscillator,
has been first introduced by Arthurs and Kelly [5]. Non-commuting system momentum p
Gauge Institute Journal Volume 6, No 1, February 2010
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and position q operators are extended to commuting operators,
θ1 = p + P
θ2 = q − Q ,
where P and Q are the momentum and position operators of an entirely independent ‘meter’
harmonic oscillator. The meter harmonic oscillator is in the vacuum state |0i with vanishing
expectation values, h0|P |0i and h0|Q|0i. The combined, system and meter, state is given by
|ψi|0i; and, the fact that [θ1 , θ2 ] is zero implies that the expectations, hψ|p|ψi and hψ|q|ψi,
can be obtained simultaneously in this scheme. From the standpoint of the original (p, q)
system, the Naimark extension provides a realization of the phase space picture of quantum
mechanics, which can be viewed as equivalent to the Schrödinger or Heisenberg pictures [6].
Theoretical constructions—such as coherent, squeezed coherent, and Bloch states—are more
naturally described in the phase space picture [7]. The experimental implementation of
squeezed coherent states requires the ancillary Hilbert spaces of a Naimark extension [8, 9].
Based on this more complete description of a particle state, Levine and Dannon [2]
have argued that the above extension represents the correct definition of system position
and momentum measurement. Examples of simultaneous measurement for non-relativistic
position and momentum, as well as spin, have been derived in earlier works by Levine and
Tucci [10, 11]. Here we explicitly derive the Naimark extension for Dirac spinors. We show
that the extension of spin and boost operators for spin-1/2 particles falls naturally into three
generations of left and right handed fermions, the starting point of the standard model for
the weak interaction [12, 13]. In addition, we suggest that left-right mixing (revealed in Z 0
couplings) and CKM inter-generational mixing (revealed in W ± couplings) are together the
phenomenology of a Lorentz group Naimark extension. The two-quark states used to derive
CKM mixings are motivated by constructions in quantum information theory.
DIRAC SPINORS AND THE LORENTZ GROUP
Consider relativistic fermions described as Dirac 4-spinors. The system is uniquely prescribed by non-commuting spin operators (notation as in Ref. [13]),


 σi 0 
Σi = 
 , i = 1, 2, 3 ,
0 σi
(1)
Gauge Institute Journal Volume 6, No 1, February 2010
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[and] the generators of infinitesimal spin rotations and boosts[,]


 a+
Si (β) = 
where a± =
q
1
(γ
2
a− σi 
a− σi
a+
,
i = 1, 2, 3 ,
(2)
√
± 1) with γ = 1/ 1 − β 2 and β = v/c. Infinitesimal boost generators are
defined in the limit β ¿ 1. To first order in β, we have
Si (β) = 1 − 21 β γ5 Σi ,
where

(3)

0 1
γ5 = 
.
1 0
(4)
The matrices γ5 Σi form a boost generator set,
{γ5 Σ1 , γ5 Σ2 , γ5 Σ3 } .
The total set of operators,
{Σ1 , Σ2 , Σ3 , γ5 Σ1 , γ5 Σ2 , γ5 Σ3 } ,
generate spin rotations and boosts that uniquely prescribe the fermion state. The condition,
[γ5 , Σi ] = 0 ,
yields commutation relations for the full operator set—given by
[Σi , Σj ] = 2i²ijk Σk ,
(5)
[Σi , γ5 Σj ] = 2i²ijk γ5 Σk ,
(6)
[γ5 Σi , γ5 Σj ] = 2i²ijk Σk ,
(7)
and
for i, j, k ∈ {1, 2, 3}. The operators and commutation relations in Eqs. (5)–(7) define the
classic spin-1/2 Lorentz group.
Gauge Institute Journal Volume 6, No 1, February 2010
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NAIMARK EXTENSION OF DIRAC SPINORS
The Naimark extension of Eqs. (5)–(7) is constructed by first defining mutually commuting pairs,
½
¾
1
(1
2
+ γ5 )Σi ,
1
(1
2
− γ5 )Σi , i = 1, 2, 3 ,
(8)
of separate left and right handed operators, and then carrying operators with different
i-values on different independent fermions. In Refs. [10] and [11], for non-relativistic position/momentum and spin, a distinction is made between quantum measurements entirely on
vacuum meters and measurements in which the original system participates. An example of
the latter case that allows simultaneity for relativistic spin and momentum reduces to the
problem of finding entangled Hilbert spaces in which non-commuting observables reside on
distinguishable quarks. The spin and momentum properties of a single quark require two
other vacuum quarks to act as the meters. The final Naimark extension is given by






1
(1
2
1
(1
2
+ γ5 )Σ1
− γ5 )Σ1
1
(1
2
1
(1
2
+ γ5 )Σ̄2
− γ5 )Σ̄2
1
(1
2
1
(1
2

¯
+ γ5 )Σ̄
3 
¯
− γ5 )Σ̄
3

,


(9)
where each component i = 1, 2, 3 is represented in a different fermionic generation. Here,
the different generations are denoted by a different number of operator overbars. Note that
the expectation values of
1
(1
2
± γ5 )Σi =
1
(1
2
± γ5 ) Σi 12 (1 ± γ5 )
(10)
for i = 1, 2, 3 are equivalent to a two-step process in which the state |ψi is projected onto
left (L) or right (R) components,
|ψiL =
1
(1
2
− γ5 ) |ψi
|ψiR =
1
(1
2
+ γ5 ) |ψi,
and the expectation value of the spin operator follows.
FERMIONIC REPRESENTATIONS OF LORENTZ NAIMARK EXTENSION AND
ENTANGLEMENTS DUE TO VECTOR BOSONS
The pattern in Eq. (9), a Naimark extension of the Lorentz group of spin and boost
operators, suggests an underlying representational structure for known massive fermions
Gauge Institute Journal Volume 6, No 1, February 2010
given by


 eR µR τR 

,
eL µL τL

(11)

 uR cR tR 

,
uL cL tL
and
6

(12)

 dR sR bR 

,
dL sL bL
(13)
with a similar pattern for left handed (massless) neutrinos.
In addition to mutual commutativity via ancillary Hilbert spaces, observable simultaneity
requires entanglement with vacuum meters. The mixings among the patterns in Eqs. (11)–
(13) are an indication that this extended quantum representation appears at the time-space
scale of the weak interaction. As described in Ref. [14], entanglement for quarks arises from
the diagonalization of the couplings to Higgs particles expressed as rotations,
f~R0 = Wu(d) · f~R
(14)
f~L0 = Uu(d) · f~L ,
(15)
and
where u(d) corresponds to up(down) quarks and
f~ = (f 1 , f 2 , f 3 ) T
corresponds to triplets (u, c, t) and (d, s, b). The CKM matrix, Uu† Ud , is the only observable
mixing across generations in the standard model—as revealed in W ± -mediated left handed
processes.
This inter-generational mixing is not observable in neutral currents coupled to Z 0 . However, in the standard model hypercharge provides a mixing of left and right handed fermions
that is observable in Z 0 coupling—but not observable in the left-isospin coupled W ± bosons.
The Z 0 boson couples to the neutral fermionic current,
j0µ = f¯γ µ (Cv − Ca γ 5 )f,
(16)
where Cv and Ca are dependent on the fermion type, and Cv further depends on the weak
angle θw ≈ 28.7o (Note that, for left handed neutrinos, there is no left-right mixing via Cv
Gauge Institute Journal Volume 6, No 1, February 2010
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and Ca ). The expression in Eq. (16) can be written as a pure vector current F̄ γ µ F , where
F = η1 fL + η2 fR
(17)
with
1q
Cv + Ca ,
X
1q
=
Cv − Ca ,
X
η1 =
η2
and
q
X =
|Cv + Ca |2 + |Cv − Ca |2 .
Equation (17) demonstrates that Z 0 coupling involves an entanglement between left and
right handed quarks. Accordingly, without Z 0 interaction, right handed quarks exist as
spectators to W ± -mediated isospin-changing transitions in a left handed quark Hilbert space.
This condition leads to the observed Cabibbo-like entanglement. For example, considering
only the first two generations of up-type (up, charm) and down-type (down, strange) quarks,
a two-quark input state to W ± and Z 0 decays is given by
|ψi = βd |dL d0R 0i + βs |sL s0R 0i
+ βu |uL u0R 0i + βc |cL c0R 0i ,
(18)
where βd2 + βs2 + βu2 + βc2 = 1, and |0i represents the vacuum in a boson Hilbert space E given
by { |Z 0 i, |W + i, |W − i, |0i }. If we assume that the W ± decays are described by a unitary
operator V , then the output state is given by
|ψ 0 i = V |ψi
= βd hdL d0R 0|V |dL d0R 0i |dL d0R 0i + βd hdL d0R Z 0 |V |dL d0R 0i|dL d0R Z 0 i
+ βs hsL s0R 0|V |sL s0R 0i |sL s0R 0i + βs hsL s0R Z 0 |V |sL s0R 0i |sL s0R Z 0 i
+ βu huL u0R 0|V |uL u0R 0i |uL u0R 0i + βu huL u0R Z 0 |V |uL u0R 0i |uL u0R Z 0 i
+ βc hcL c0R 0|V |cL c0R 0i |cL c0R 0i + βc hcL c0R Z 0 |V |cL c0R 0i |cL c0R Z 0 i
+ βd ( huL W − |V |dL 0i |uL i + hcL W − |V |dL 0i |cL i ) |d0R W − i
+ βs ( huL W − |V |sL 0i |uL i + hcL W − |V |sL 0i |cL i ) |s0R W − i
+ βu ( hdL W + |V |uL 0i |dL i + hsL W + |V |uL 0i |sL i ) |u0R W + i
+ βc ( hdL W + |V |cL 0i |dL i + hsL W + |V |cL 0i |sL i ) |c0R W + i ,
Gauge Institute Journal Volume 6, No 1, February 2010
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where the Z 0 boson couples to either qL or qR0 and the W ± couples only to qL . For
notational clarity we have dropped the qR0 dependence in W ± -mediated decays—i.e.,
hqL qR0 W ± |V |q̃L qR0 0i −→ hqL W ± |V |q̃L 0i. Unitarity of V implies the following conditions:
| huL W − |V |dL 0i |2 + | hcL W − |V |dL 0i |2 + | hdL d0R Z 0 |V |dL d0R 0i |2 + | hdL d0R 0|V |dL d0R 0i |2 =
| huL W − |V |sL 0i |2 + | hcL W − |V |sL 0i |2 + | hsL s0R Z 0 |V |sL s0R 0i |2 + | hsL s0R 0|V |sL s0R 0i |2 =
| hdL W + |V |uL 0i |2 + | hsL W + |V |uL 0i |2 + | huL u0R Z 0 |V |uL u0R 0i |2 + | huL u0R 0|V |uL u0R 0i |2 =
| hdL W + |V |cL 0i |2 + | hsL W + |V |cL 0i |2 + | hcL c0R Z 0 |V |cL c0R 0i |2 + | hcL c0R 0|V |cL c0R 0i |2 =
1.0,
(19)
huL W − |V |dL 0i∗ huL W − |V |sL 0i + hcL W − |V |dL 0i∗ hcL W − |V |sL 0i = 0 ,
(20)
hdL W + |V |uL 0i∗ hdL W + |V |cL 0i + hsL W + |V |wL )i∗ hsL W + |V |cL 0i = 0 .
(21)
and
The output density matrix ρ0 for the left handed system is obtained by taking the trace of
the right handed quark and vector boson Hilbert spaces, R and E, respectively. This gives
the result,
ρ0 = T rRE ( |ψ 0 ihψ 0 | )
(22)
= βd2 A2d |dL ihdL | + βs2 A2s |sL ihsL |
+ βu2 (1 − A2u ) |ψu ihψu | + βc2 (1 − A2c ) |ψc ihψc |
+ βu2 A2u |uL ihuL | + βc2 A2c |cL ihcL |
+ βd2 (1 − A2d ) |ψd ihψd | + βs2 (1 − A2s ) |ψs ihψs | ,
(23)
where
A2q = | hqL qR0 0|V |qL qR0 0i |2 + | hqL qR0 Z 0 |V |qL qR0 0i |2 , q = d, s, u, c ,
(24)
and
|ψd i = q
1
1−
A2d
( huL W − |V |dL 0i|uL i + hcL W − |V |dL 0i|cL i ) ,
(25)
Gauge Institute Journal Volume 6, No 1, February 2010
|ψs i = q
|ψu i = q
|ψc i = q
1
1−
1
A2s
1 − A2u
1
1−
A2c
9
( huL W − |V |sL 0i|uL i + hcL W − |V |sL 0i|cL i ) ,
(26)
( hdL W + |V |uL 0i|dL i + hsL W + |V |uL 0i|sL i ) ,
(27)
( hdL W + |V |cL 0i|dL i + hsL W + |V |cL 0i|sL i ) .
(28)
From the unitarity conditions in Eqs. (20) and (21), we have hψd |ψs i = 0 and hψu |ψc i = 0.
By considering the W ± -mediated decays of left handed quarks with right handed quark
spectators, orthogonal bases { |ψd i, |ψs i } and { |ψu i, |ψc i } emerge that are rotated relative
to sets { |uL i, |cL i } and { |dL i, |sL i }, respectively.
It should be emphasized that Z 0 left-right mixing does not appear in the current F̄ γµ F .
Consequently, although the right handed spectators and left handed system quarks in
Eq. (18) both interact with the Z 0 boson, they do not mix handedness. Furthermore,
the W ± channel is isolated to left handed inputs. Even without the Z 0 boson, quark mass
is a left-right mixing parameter for the Naimark extension in Eq. (9). However, we are motivated by the idea that the weak bosons represent a dynamical entangling mechanism for a
Lorentz group Naimark extension. The left-right mixing in Eq. (17) combine with the CKM
matrix for a full Naimark extension of the Lorentz group of spin and boost operators. The
mixing involves the weak angle θw and the CKM matrix parameter set (most prominently
the Cabibbo angle θC ≈ 13.1o ).
QUANTUM INFORMATION
In quantum information theory the state |ψi in Eq. (18) is known as a purification of the
input density matrix ρ = T raceRE (|ψi hψ|) [15]. The concept of purification is at the center
of definitions of entanglement fidelity and entropy exchange for transmission through noisy
quantum channels [16–18]. In a follow-on paper we evaluate the entanglement in Eqs. (25)–
(28) by considering the W ± -mediated weak decay as a noisy quantum information channel
[19]. Weak decays of the four left handed fermions (u, d, c, s) are interpreted as a single
quantum information channel, which we denote the W ± channel. The noise in the channel
results from weak boson interactions. This interpretation of particle decays as information
Gauge Institute Journal Volume 6, No 1, February 2010
10
channels also extends the theory of quantum measurement. A generalization of quantum
statistical counting to accessibly distinguishable particles results in an entropic criterion for
quark mixing angles equivalent to the GIM mechanism [20].
This work was supported in part by the STFC Grant PP/C50209X/1.
∗
Electronic address: [email protected]; corresponding author.
†
Electronic address: [email protected]
[1] S. Weinberg, Quantum Theory of Fields, Volume 1 (Cambridge University Press, Cambridge,
1995).
[2] R. Y. Levine and H. V. Dannon, Gauge Inst. J. of Math. & Phys. 1, 2 (2005).
[3] M. A. Naimark, Izv. Akad. Nauk. SSSR, Ser. Mat. 4, 227 (1940).
[4] M. A. Naimark, C. R. Acad. Sci. URSS 41, 359 (1943).
[5] E. Arthurs and J. L. Kelly, Bell Syst. Tech. J. 44, 725 (1965).
[6] Y. S. Kim and E. P. Wigner, Am. J. Phys. 58, 439, (1990).
[7] J. R. Klauder and B.-S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics, (World Scientific, Singapore, 1985).
[8] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55,
2409, (1985).
[9] H. P. Yuen, Phys. Rev. A13, 2226, (1976).
[10] R. Y. Levine and R. R. Tucci, Found. Phys. 19, 161 (1989).
[11] R. Y. Levine and R. R. Tucci, Found. Phys. 19, 175 (1989).
[12] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
[13] D. Griffiths, Introduction to Elementary Particles (J. Wiley and Sons, New York, 1987).
[14] M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley,
New York, 1995).
[15] L. P. Hughston, R. Jozsa, and W. K. Wooters, Phys. Lett. A183, 14 (1993).
[16] B. Schumacher, Phys. Rev. A54, 2614 (1996).
[17] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wooters,
Phys. Rev. Lett. 76, 722 (1996).
Gauge Institute Journal Volume 6, No 1, February 2010
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[18] B. Schumacher and M. A. Nielsen, Phys. Rev. A54, 2629 (1996).
[19] R. Y. Levine and J. Y-K. Cho, ’The W ± -Mediated Weak Decay as an Information Channel,’
(in preparation).
[20] S. Glashow, J. Illiopolous, and L. Maiani, Phys. Rev. D2, 1285 (1970).
The 9 -Mediated Weak Decay as an
Information Channel
Robert Y. Levine
Spectral Sciences, Inc. ,
Burlington, MA, USA
bob@spectral. com; Corresponding Author
James Y-K. Cho
School of Mathematical Sciences,
Queen Mary, University of London,
London, UK
J. Cho@qmu. ac. uk
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< uW V s > u > < cW V s > c > s W >
< dW V u > d > < sW V u > s > u W >
< dW V c > d > < sW V c > s > c W > ;
1
2
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2
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2
< ss V ss > < sW V c >
: ;
2
+
2
< cW V d >
+
2
we have the conditions
< sW V u >
2
0
+
V d > V dd >
V s > V ss >
V u > V uu >
V c > V cc >
2
V
( 5)
2
< uu V uu > 2
2
2
2
< cc V cc > 2
( 6)
< uW V d > < uW V s > < cW V d > < cW V s > ;
( 7)
and
< dW V u > < dW V c > < sW V u > < sW V c > :
+
+
+
+
( 8)
For notational clarity, b ecause the channel operation with W output does not involve right handed quarks, the q state is dropped
< qW V q > .
from the matrix element; < qq W V q q >
The conditions in Eqs. ( 6) -( 8) are equivalent to the W information channel having ` trace preserving operator elements' [ 1 4] . Equations ( 7) and ( 8) insure that the ratio of C abibb o-allowed to C abibb osuppressed channel amplitudes is the same for each of the up-type
and down-type quarks ( although not necessarily that the up-type
and down-type ratios are equal) . This fact further motivates the
trace preserving channel representation for these decays.
Gauge Institute Journal Volume 6, No 1 , February 201 0
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;
d; s ; u; c @ >=
1 A 2@ d > < d + 2 A 2I s > < s + 3 ( 1 A 2K ) K > < K +
4 ( 1 A 3? ) ? > < ? + 3 A 2K u > < u + 4 A 2? c > < c +
1 ( 1 A 2@ ) @ > < @ + 2 ( 1 A 2I ) I > < I ;
q
@ I > = 0;
K ? > = 0:
< cW V d0 > c > ;
< cW V s 0 > c > ;
< s W + V u0 > s > ;
!
< s W + V c0 > s > :
"
&
$
' $ () )* '
) * K > ; ? > @ > ; I > * d > ; s > u > ; c > )* +
Q R '
E ) <
Robert Y. Levine and James Y-K Cho
'
W
< qW V q >
W Z
V
9 C hannel C apacity
3
! "# $# % &# % '( ! % &(
) * % + ,
% + n # "+ n n
n!
n
"
n
d >< d
c >< c
:
s >< s
u >< u
* % -+
# . / 0 "# / W
W
Z
S S d >< d S s >< s
S u >< u S c >< c ;
* % 1+
S : * "2+
S Trace ; i
; : : : ; ! % &( 3
) * % 2+ s >; d > u >; c >
E
!
"
i
i
i i ) * % 1+ # sd uc * "% +
Gauge Institute Journal Volume 6, No 1 , February 201 0
@I (
3 ( 1
4 ( 1
1 2@ + 2 2I +
K ) ( co s + sin ) ( co s + sin ) +
2? ) ( sin + cos ) ( sin + cos ) ) =
2
K? (
1 ( 1
2 ( 1
K? (
)
3 2K + 4 2? +
@ ) ( co s + sin ) ( co s + sin ) +
2I ) ( sin + cos ) ( sin + cos ) ) =
2
! ! "
F F G
cos
=
+ 0 1
sin
=
+
1
cos
=
0
0
1
G
2
#
2
K
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@
%
K
2
@
&
& ' ( ) sin
=
0 Ò =
@I
=
1 2@ + 2 2I
= (
3 ( 1
+
Ò)
2
'
2
K
2
1
) +
4 ( 1
?
2
)
1 2@ 2 2I ) 2 + ( 3 ( 1 2K ) 4 ( 1 2? ) ) 2 +
2
2 ( 1 2 2I ) ( 3 ( 1 2K ) 4 ( 1 2? ) ) cos 2 @
(
, = (
Ê)
2
*
+
! Robert Y. Levine and James Y-K Cho
3 2K
3 2K
3 2K
4 2? 1 2@ 2 2I 4 2? 2 1 2@ 2 2I 2 2
2
2
4 ? 1 @
2 I K? + + 2
2
+ + 2
2
2 2
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( %#!
# "$ ## ' ')
'*
% E +)
! ')
"' 4 !
4 + 2 2 2I + 1 2 2@ 2
3 2 K 4 2 2? ,
Gauge Institute Journal Volume 6, No 1 , February 201 0
4
1
0
+
2
+
3
:
4
1
4(
+
4
0
2 ( )
= 1
2
3
4)
=
E
ln =
)
(1
E
ln
E
) =1
ln( 1
C hannel C apacity in the High Energy Limit
! " ##"
$
# #" " % # &#
$ "
=
K
+ 6
(
) =
2) ( +
( 1 2) ( +
2
2( ) (1
0
F
F
=
= (1
@
=
+
ln( +
Ò ) ln(
Ê)
I
F
F
=
Ò)
Ê)
2) ( ( 1 2) ( (1
F
F
Ê)
+ (1
1 2 ) 2
+
1 2) ( 1
2 ) 1 2( 1
2 ) Ò =
234 4
+
21 2 ( 1
2 ) 2
+ 2
34 1 2 2 ( 1
2 )
cos 2
Ê =
21 2 4
+
234 ( 1
2 ) 2
+ 2
1 2 34 2 ( 1
2)
cos 2
12
=
ln( Ò ) ln(
F
F
1 2 2
=
?
1
+
2
3
= 1
'
(
)
*
4
34 3 4 '
+
" 1 2 , 1 2
=
1
2 =
Ò
Ê
Ò)
Ê)
Robert Y. Levine and James Y-K Cho
!
!" ! " #" $#" !$ % & !$ '
!
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(
" )
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! " + $$ ! (, (- "
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& $
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& $ !$ % 1*2 ! , . + 5
C onclusions
* $
) $ " $" #
$ ) $
0$
#
3#
$ " " !$ $ Gauge Institute Journal Volume 6, No 1 , February 201 0
"
" $ # # 34- % & ' # ! 9
( " # Robert Y. Levine and James Y-K Cho
#
A Reversibility of the 9 Decay Channel
E - = Trace 43 (
><
)
!"#
> $ !# $
R Q %
- = ( B@ > < B@ + ( 1
( BI > < BI + ( 1
( BK > < BK + ( 1
( B? > < B? + ( 1
1
2
3
4
A@) W
AI ) W
AK) W
A? ) W
2
2
2
2
+
+
>< W
>< W
>< W
>< W
+
+
)+
)+
)+
);
!&#
A G ' q = d; s; u; c' $ ( ( # BG > ' q = d; s; u; c'
0 > ; Z > 0
BG > = < qq 0 V qq 0 > 0 > + < qq Z V qq 0 > Z > :
0
!)#
0
*
< qq Z V qq 0 > Z q q q
q W
0
0
Irr ( ; ) = S ( ) + S ( - ) S ( ( ) ) ;
+#
( ) $ ( # ,( # ,-# ' Irr ( ; ) ( .
Gauge Institute Journal Volume 6, No 1 , February 201 0
$
References
[1 ]
C . E.
S hanno n
and
C o mmunic at i o n,
[2]
A. S .
[3]
B.
H o le vo ,
[ 4]
R . Y.
L e v ine
L . P.
H ughs t o n,
( 1 993 )
[ 7]
R . Y.
( 3)
[ 8]
and
M ay,
B . W.
and
J.
I nf.
P re s s ,
""
T he o ry,
T he o ry
U rb ana,
( 1 998)
We s t mo rel and ,
` E xt ens io ns
Fe rmio ni c
of
1 949) .
3 69.
P hy s .
)#$
Rev.
o f t he
L o re nt z
R e pre s ent at io ns ' ,
G ro up
to
be
and
s ub mit -
2 008.
R.
J o z s a,
and W. K .
Wo o t t ers ,
P hy s .
L et t .
) &!
1 4.
S chumache r,
L e v ine ,
( A ugus t ,
B . W.
M at hemat i cal
o f I llino is
M. D.
C ho ,
H and e d
t ed ,
[ 6]
Trans .
T he
1 31 .
L e ft - R ight
[5]
We ave r,
( U nive rs i ty
IEEE
S chumache r
( 1 997)
W.
P hy s .
G auge
)#"
Rev.
I ns t it ut e
( 1 996)
J o urnal
of
2 61 4.
M at h
and
P hy s ic s
2 006) .
S chumache r
and
M. A.
N ie ls e n,
P hy s .
Rev.
)#"
( 1 996)
2 62 9.
[ 9]
C . H.
B e nne t t ,
S mo lin,
[ 1 0]
[1 1 ]
N.
S.
G.
B ras s ard ,
and
W. K .
C ab ib b o ,
P hy s
G las how,
J.
Wo o t t e rs ,
Rev
L et t
I l lio p o lo us ,
S.
P o p e s c u,
P hy s .
Rev.
( 1 963 )
and L .
B.
S chumache r,
L et t .
%$
( 1 996)
J. A.
72 2 .
531 .
M aiani,
P hy s .
R ev .
,
( 1 970)
1 2 85 .
[1 2]
N.
K o b ayas hi
and
T.
M as kawa,
P ro g.
T he o r.
P hy s .
"'
( 1 973 )
65 2 .
[1 3]
M.
P e s k in and D .
T he o ry,
S chro e d e r,
( A d d is o n- We s le y
A n I nt ro d uc t io n t o Q uant um F ie ld
P ub lis hing
C o mpany,
N ew
Yo rk ,
1 995 ) .
[ 1 4]
M. A.
N ie ls e n
Q uant um
b rid ge ,
[1 5]
S.
and
I . L.
I nfo rmat io n,
C huang,
Q uant um
( C amb rid ge
U nive rs i ty
2 000) .
L loy d ,
P hy s .
Rev.
)#$
( 1 997)
C o mput at i o n
1 61 3 .
P res s ,
and
C am-