The spacetime attribute of matter

6 Pages·0.11 MB·English

by  Borge Nodland

The spacetime attribute of matter
Borge Nodland
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627
We propose that spacetime is fundamentally a property of matter, inseparable from it. This leads us to suggest that
all properties of matter must be elevated to the same status as that of spacetime in quantum field theories of
matter. We suggest a specific method for extending field theories to accomodate this, and point out how this leads
to the evolution of fields through channels other than the spacetime channel.
PACS numbers: 11.10.-z, 03.65.Bz, 03.65.-w
(December, 1997)
I. INTRODUCTION
8
9 In general,an interacting quantum mechanical system is described by an operator equation for the field amplitude
9 f of the system. This field equation is of the form
1
∂
n O( ,x,a )f(x)=I (x) (1)
a ∂x i int
J
(throughoutthis paper,weuse anoverhattobdenoteoperators). Thespacetime-dependenttermI (x) in(1)is given
3 int
by

2
∂L
v I (x)= int, (2)
0 int ∂f
5
2 and arises from the Euler-Lagrangeequation
2
1 ∂L ∂L
−∂ [ ]=0 (3)
7 ∂f µ ∂(∂ f)
µ
9
/ when interactions are present in the form of an interaction term L in the Lagrangiandensity L [1].
h int
t Wenotethatspacetimex=xµ typicallyenterasaderivative ∂ (andalsopossiblyasaparameter)intheoperator
- ∂x
p O in (1), while other properties, or attributes, a , such as mass, spin, etc., generally enter only as parameters, not
i
e derivatives (an example is the Dirac equation). Spacetime, therefore, assumes a special role in (1).
h
bHowever, one may take the view that the spacetime location x of a particle represented by a field f is just one of
:
v many attributes the field can have. A separation of spacetime from objects or fields is artificial, since space really
i has no meaning withoutphysicalobjects init, andtime has no meaning unless a physicalchangeoccurs inanobject.
X
From these considerations, we propose that spacetime x is a regular property of matter, and we take this as an
r
a indicationthat otherpropertiesof matter shouldplaya roleofthe samesignificance asthat ofspacetime inquantum
fieldtheories. Inotherwords,spacetimeexemplifies howotherattributes ofmattershouldbe viewedinfieldtheories.
A simple, natural approach toward the attainment of a satisfactory quantum theory of gravitation is to regard
spacetime x as a discrete, quantized entity (see for example Ref. [2] and references therein). This accords with the
quantal nature of other particle properties like spin, mass, etc, and supports the notion that spacetime is a property
of matter.
An attribute of a field can exist only if the field exists. For example, since the collection of all measurable values
for the spacetime locationofa particle representedby the fieldf is just spacetime itself, spacetime exists only becuse
objects, or fields, exist. The same reasoning can be applied to other field attributes. A field may thus be considered
to carry with it severalattributive spaces, such as spacetime, a mass–space, a spin–space, etc.
The presence of spacetime derivatives in quantum field theory (i.e. in Lagrangian densities, and consequently in
generatingfunctionals,fieldequations,etc.) leadstoanevolutionofafieldamplitudeinspaceandtimeviainteractions
withotherfields. Interactionsbetweenfieldsofspecificmasses,spins,andspatiotemporallocationprobabilitiesresult
in new fields with new values for these properties. Commonly, such interactions are thought to proceed through
space and time. However,if one considers spacetime to be one of many attributive spaces carriedby a field, one may
consider interactions to go through alternative channels. A specific such channel would then be an attributive space
existing by virtue of characterizing a field. The totality of measurable values of the specific attribute of the field is
what constitutes the space or channel. Examples of such alternative channels are a mass (or energy) channel, and a
spin channel.
1
II. GENERAL APPROACH
In view of the discussion in the introduction, we now extend the interpretation of the field f to that of a function
f of several attributes (in this paper, we use a tilde throughout to denote extensions of operators and fields). This
means that f is an attributive field defined on a multiattributive space S, containing as subspaces the attributive
sepaces mentioned in the introduction (among others). f is now a multiparticle field that contains information on the
evolution (viaeinteractions) of particles into new particles characterizedby new attributive values for their spacetime
location, mass, spin, etc. e
With this extension of f in (1), the operator O in (1) must similarly be extended to an operator O that takes into
accountthe other attributes f has in the same manner that the spacetime attribute x is normally takeninto account
in(1). Thisimpliesthatanattributeai shouldnbotmerelybe aparameterin(1),butshouldalsoenteebrasaderivative
∂ .
∂ai
In view of the above, we therefore extend (1) to the general form
∂ ∂
O( , ,··· ,a ,a ,···)f(a ,a ,···)=0, (4)
∂a ∂a 1 2 1 2
1 2
eb e
where a (i=,1,2,···) are the attributes of f, including the spacetime attribute x.
i
Since f isinherentlyafieldofmultiple, andthus interacting,particles,the operatorO isassumedtoaccountforall
e
interactions. This obviatesthe need for a separateinteractionterm [as in(1)] on the righthandside of(4). One may
considerethefieldequation(1)tobeaprojectionontothespacetimesubspaceofS ofanebunderlyingfieldequation(4),
where in this projectiononly a single attributive derivative, that is, the spacetime derivative ∂ = ∂ = ∂ =∂ is
∂a1 ∂x ∂xµ µ
retained.
Inviewofthe discretenatureofanattributea (including x,asexplainedabove),the attributivespaceS isstrictly
i
speaking a discrete space over which a difference operation (or possibly some alternative operation) on f would be
more appropriate than differentiation. However, for simplicity, we still retain the notation in terms of differentiation
as in (1), with the understanding that it should be re–interpreted in terms of a difference operation, conseistent with
the quantal nature of S. We also assume that this operation is linear.
In what follows, we want to utilize specific, established expressions for the operator O( ∂ ,x,b) in (1) to obtain
∂x
information on the nature of an attribute b that appears as a derivative in (4), while only entering as a parameter in
(1) [i.e. b 6= x]. One way to do this is to apply a differential operator ∂∂b to (1). Thus, ifb(1) [with ai replaced by b,
and f(x) replaced by f(x,b)] holds, then
∂ ∂
e [O( ,x,b)f(x,b)]=0 (5)
∂b ∂x
must hold. It is seen that (5) is a two-attributebversion of (e4),
∂ ∂
O( , ,x,b)f(x,b)=0, (6)
∂x ∂b
if we extend the interpretation of the field feb(x) in (1) to theat of an attributive field f(x,b) over x and b.
e
III. APPLICATION TO THE DIRAC EQUATION
As a specific application of the above discussion, we focus on the Dirac equation
(iγµ∂ −m)ψ(x)=I (x), (7)
µ int
where γµ (µ = 0,1,2,3) are the Dirac gamma matrices, m the mass of ψ, and ∂ = ∂ the spacetime derivative
µ ∂x
operator. From(7),weseethatexamplesofthenon–spacetimeattributesa in(1)arequantitieslikem(representing
i
mass) and γµ (representing spin).
According to the discussionin the previous section, we now extend the interpretationof the Dirac field ψ(x) in (7)
to that of a field ψ(x,m) over spacetime x and mass m. We next differentiate (7) [with ψ(x) replaced by ψ(x,m)]
with respect to mass m, corresponding to the operation in (5), with the attribute b of the general field f being the
mass attribute m oef the extended Dirac field ψ, e
e
2
∂ ∂
(iγµ∂ −m)ψ(x,m)= I (x). (8)
∂m µ ∂m int
This results in the equation e
∂ψ
(iγµ∂ −m) −ψ =0. (9)
µ
∂m
e
To solve this equation, we note that the Dirac operator e
D(x,m)=(iγµ∂ −m) (10)
µ
has the inverse (i.e. the corresponding Dirac propagator)
b
S(x,m)=(iγµ∂ +m)∆ (x,m), (11)
µ F
since
D(x,m)S(x,m)
=(iγµ∂ +m)(iγν∂ +m)∆ (x,m)
µ ν F
=(b−2−m2)∆ (x,m)=δ4(x). (12)
F
Here ∆ (x,m) is the Feynman propagator, 2 the contraction ∂µ∂ , and δ(x) the four–dimensional delta function.
F µ
(12) follows from the symmetry property ∂ ∂ =∂ ∂ of the derivative operator ∂ ∂ , the anticommutator property
µ ν ν µ µ ν
{γµ,γν}=2gµν [where gµν is the metric of signature (1, -1, -1, -1)] of the gamma matrices γµ, and the fact that the
Feynman propagatoris the inverse of the Klein–Gordonoperator K(x,m),
K(x,m)∆ (x,m)=(2+m2)∆ (x,m)
F b F
=−δ4(x). (13)
b
In accord with the general discussion in the previous section, we have included here the specific m–dependence in
all operators and propagators. As with the general operator O, we assume that its propagator inverse O−1 has an
−1 −1
extension O that is generally different from its spacetime projection (i.e. O 6=O−1).
b b
Multiplication of S(x−x ,m) on both sides of (9), and integrating with respect to spacetime x yields, with the
1 1
use of (12)eb, eb b
∂ψ(x,m)
= S(x−x ,m)ψ(x ,m)d4x . (14)
∂m 1 1 1
Z
e
We see that the integro-differentialequation (14)indicates thateψ undergoesan interaction–drivenevolutionthrough
a mass channel, as discussed in the introduction. Consequently,we define a mass evolution operator M(x,m,m ) by
0
e
ψ(x,m)=M(x,m,m0)ψ(x,m0), c (15)
where ψ(x,m ) is an extended Dirac field characterizing a system with a fixed mass m . Clearly, we have
0 e c e 0
e M(x,m0,m0)=1. (16)
In view of the definition (15) for M, (14) is ecquivalent to
∂M(x,m,m )
c 0 = S(x−x ,m)M(x ,m,m )d4x . (17)
∂m 1 1 0 1
Z
c
But the differential equation (17) and the boundary conditicon (16) can be combined to give the single integral
equation [with x replaced by x ]
0
M(x ,m,m )
0 0
m
=1+ dm d4x S(x −x ,m )
c 1 1 0 1 1
Zm0 Z
M(x ,m ,m ). (18)
1 1 0
c
3
Iteration of (18) yields the perturbation solution
m
M(x ,m,m )=1+ dm d4x S(x −x ,m )
0 0 1 1 0 1 1
Zm0 Z
m m1
+ c dm d4x S(x −x ,m ) dm d4x
1 1 0 1 1 2 2
Zm0 Z Zm0 Z
S(x −x ,m ) + ···
1 2 2
m m1
+ dm d4x S(x −x ,m ) dm d4x
1 1 0 1 1 2 2
Zm0 Z Zm0 Z
mn−1
S(x −x ,m ) ··· dm d4x
1 2 2 n n
Zm0 Z
S(xn−1−xn,mn) + ··· (19)
It is clear from the integration sequences in (19) that m1 ≥m2 ≥ ··· ≥ mn−1 ≥mn. Therefore, each product of
Dirac propagatorsS in (19) is in mass orderedform (which is analogous to the time ordering of the various products
of Hamiltonian operators in the time evolution operator U(t,t ) [3]). We now define a mass ordering operator M for
0
the product of two quantities A (m ) and A (m ) by
1 1 2 2
b c
A (m )A (m ) if m ≥m
M[A (m )A (m )]= 1 1 2 2 1 2 (20)
1 1 2 2 A (m )A (m ) if m <m .
2 2 1 1 1 2
(cid:26)
c
M is defined in a similar manner for products of more than two quantities A (m ). By using M to retain the mass
i i
ordering in (19), we can now extend the mass integrations in the various terms in (19) to the full range from m to
0
mcby including an 1 compensation factor for the n–th term. Thus, c
n!
∞ 1 m m
M(x ,m,m )=1+ dm dm ···
0 0 n! 1 2
n=1 Zm0 Zm0
X
cm
dm ··· d4x d4x ··· d4x
n 1 2 n
Zm0 Z Z Z
M[S(x −x ,m )S(x −x ,m ) ···
0 1 1 1 2 2
S(xn−1−xn,mn)]. (21)
c
Since (21) has the form of a Taylor series expansion of the exponential function, we write M(x,m,m ) on the
0
shorthand form
c
m
M(x,m,m )=Mexp dm′ d4x′S(x−x′,m′) . (22)
0
Zm0 Z !
c c
It is seen that the above derivation of the expression for M(x,m,m ) is similar to the derivation of the expression
0
for the standard time evolution operator U(t,t ) [3,4].
0
For definiteness, we represent the Feynman propagator bcy the four–dimensional Fourier space integral solution of
(13) [1], b
1 e−ikx
∆ (x,m)= d4k, (23)
F (2π)4 k2−m2
Z
where k is the four–dimensional wave vector
k =kµ =(k ,~k). (24)
0
For simplicity of notation, we here assume that the k–integration involves the proper path around the two poles on
the k axis [thus omitting the usual infinitesimal parameter ǫ in the denominator of (23)] [1]. Substitution of (23)
0
into (11) yields
1 (γµk +m)e−ikx
S(x,m)= µ d4k. (25)
(2π)4 k2−m2
Z
4
One can now substitute (25) into (21) and carry out the mass integrations at the various orders in (21). To first
order we have
m
M1(x ,m,m )= dm d4x S(x −x ,m )
0 0 1 1 0 1 1
Zm0 Z
= c1 mdm d4x d4k(γµkµ+m1)e−ik(x0−x1)
(2π)4 1 1 k2−m2
Zm0 Z Z 1
= 1 d4k d4x γµkµ tanh−1 m
(2π)4 1 |k| |k|
"
Z Z
−tanh−1 m0 + 1lnm20−k2 e−ik(x0−x1), (26)
|k| 2 m2−k2
! #
where |k|= k2−~k2.
0
In accordqwith the previous discussion, we view the propagation of the attributive field ψ(x ,m ) in (9) as taking
0 0
place throughboth a spacetime channeland a mass channel. To propagate ψ(x ,m ) to ψ(x,m), one may propagate
0 0
the field through spacetime only by use of an extended Dirac propagator S(x,m,m ) thaet contains in it the mass
0
channel propagation from ψ(x ,m ) to ψ(x ,m). From (26), we see thatethe first-ordeer term of a perturbation
0 0 0
expansion of S(x,m,m ) has the Fourier space representation e
0
e e
S1(x,m,m )
e 0
= e 1 γµkµ tanh−1 m −tanh−1 m0
(2π)4 |k| |k| |k|
" !
Z
1 m2−k2
+ ln 0 e−ikxd4k. (27)
2 m2−k2
#
Higher order contributions to the extended propagator S(x,m,m ) can readily be computed from (25) and (21).
0
A procedure similar to that used for the mass attribute of the extended Dirac field ψ can be applied for the spin
attribute of ψ. In analogywith the case above involvingemass m, the point of departure is to interpret the quantities
γµ in (7) to represent the spin of ψ. We use the metric gµν of signature (1,−1,−1,−1)eto write (7) on the form
e
(igµλγ ∂ −m)ψ(x)=I (x), (28)
λ µ int
e
and then differentiate (28) with respect to γ . The result is [with the replacement of ψ(x) by the extended field
σ
ψ(x,γσ)]
∂ψ(x,γσ) ∂ψ(x,γσ)
e igµσ∂ ψ(x,γσ)+iγµ∂ −m =0 (29)
µ µ
∂γ ∂γ
σ σ
or e e
e
∂ψ
(iγµ∂ −m) +i∂σψ =0. (30)
µ
∂γ
σ
e
The resulting integro-differentialequation [analogous to (14)] neow becomes
∂ψ(x,γσ)
i = S(x−x ,m)∂σψ(x ,γσ)d4x , (31)
∂γ 1 1 1
σ
Z
e
whichmaybeinterpretedasdescribinganevolutionofψ througheaspinchannel. Furtheranalysisforeachcomponent
of γσ can be carried out straightforwardlyin a manner analogous to the one above for m.
It is interesting to note that the attributive approaceh taken here, which is based on the Dirac equation, leads to
a specific relationship between the spin-space and mass-space evolution of the extended Dirac field. From (31) and
(14) we see that these non-spacetime evolutions are connected via the operational relation
∂ ∂ ∂
i = (32)
∂γ ∂m∂x
σ σ
between the spin (γσ), mass (m), and spacetime (xσ) attributes of the extended Dirac field.
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IV. SUMMARY AND CONCLUSIONS
We have proposed that the spacetime location x of particles represented by a field f is nothing more than an
attribute of the field itself. This obviates the need for treating space and time in any special manner, as is done in
fieldtheoriesofmatterinparticular,andphysicsingeneral. Asapointofdeparturecompaetiblewiththisproposal,we
havetreated other attributes of the field ina wayequivalentto the currenttreatmentofspacetime in field theoriesof
matter. Specifically,wehavesuggestedhowfieldequationsmaybeextendedwhenelevatingnon-spacetimeattributes
of a field to the same status as that of the spacetime attribute. We finally applied this to the case of an attributive
extension of the Dirac equation.
ACKNOWLEDGMENTS
This researchwas supported by NSF Grant No. PHY94-15583.
[1] L. H. Ryder,Quantum Field Theory (Cambridge University Press, Cambridge, 1985).
[2] G. ’t Hooft, Class. QuantumGrav. 13, 1023 (1996).
[3] J. D.Bjorken and S.D. Drell, Relativistic Quantum Fields (McGraw–Hill, New York,1965).
[4] J. J. Sakurai, Advanced Quantum Mechanics (Addison–Wesley, Reading, 1967).
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