Theory of Quantum Gravity of photon confirms
experimental results of a varying fine structure constant while Quantum Mechanics leads to
String theory
Pradip Kumar Chatterjee
Indian Physical Society
IACS Campus,2A & 2B Raja Subodh Chandra Mullick Road,Calcutta 700032
India
Abstract
Quantum Mechanics of photons leads to a theory of Quantum Gravity
that nicely matches the experimental results of varying fine structure
constant, obtained from many-multiplet Quaser absorption systems and
atomic clocks. The variation of that constant is due to quantum gravity of
photons, created by their non-zero invariant mass. The photon mass is
obtained from a Klein-Gordon scalar tachyon. This led to a Lorentz
symmetry-breaking and varying speed of light theory in complex
spacetime manifold. In essence, Quantum Mechanics includes quantum
gravitational potential in the guise of Quantum potential. The greatest
surprise lies in showing that Quantum Mechanics naturally leads to open
bosonic string whose troublesome tachyonic vibration is taken in its
stride. Quantum Mechanics also proves Sen`s second conjecture and
space-tearing. Length melts into dimensionless number at the Planck
scale. Quantum-mechanical analog of the classical equation E=Mc2has
been derived and a dispersion relation demonstrates Lorentz non-
invariance in Quantum Mechanics.
1. INTRODUCTION
Theory of Quantum Gravity is perhaps the greatest intellectual challenge to physicists today. Although the three interactions in nature --- electromagnetic, weak and strong --- have been successfully described by Quantum Field Theories (that constitute the Standard Model), gravity has
persistently resisted unification with Quantum theory. There are several distinct paths along which attempts have been made in the past [1], but the following two theories have emerged with a sustained growth and internal consistency; and their recent results have far-reaching consequences in the understanding of our physical world:
(1) Loop Quantum Gravity [2,3,4,5,6,7]
(2) String, Superstring, M-theory [8,9,10,11]
It is a real conundrum why gravity cannot be united with Quantum Mechanics.
When Quantum Gravity is teased out of Quantum Mechanics in a deceptively
simple way, its mathematics spontaneously leads to the theory of open strings [12] and also tachyon condensation [13], when photon dynamics is considered.
It is really happenstance that quantum gravity has found its experimental verification in the recent diverse experiments aimed at finding variations of fine
structure constant over different look-back times [14,15,16].The first hint of a
varying fine structure constant surfaced in string theory. Since then there has been a surge of interest in finding observational and theoretical results in
support of a varying α. In 1999, J.K.Webb et al [14] declared results showing a time-varying α. Peres [17] suggested that the variation in α is due to a varying speed of light (VSL). The controversy regarding the effect of a varying
dimensionful quantity on a dimensionless constant has been discussed below,
and a solution to this end emerged when one could identify quantum spacetime with complex spacetime manifold. The main purpose of this paper is to show that our theoretical predictions from the quantum gravity of photons agree quite nicely with the accumulated experimental results of a time-varying α. The other aim is to access string theory through Quantum Mechanics.
Various theories have been advanced to interpret a varying αover cosmological times [18,19,20,21,22]. A varying αcreates many troubles in present-day physics ---- chief among these being a contradiction of the Standard Model of particle physics. It however more than compensates for this by yielding a theory of quantum gravity of photons, nicely verified by the extant experimental results provided by many-multiplet quaser absorption systems, Gamma Ray Bursts(GRB) and transitions between two nearly degenerate states of atomic clock [14,15,16,23,24,25,26,27,28,29,30,31,32,33,34].
Another striking result is obtained from this quantum gravity theory in a form of mathematical gift : Quantum Mechanics,if probed deeply, leads to open string theory [12]. A three-dimensional vibration also requires the use of quaternions in Quantum theory [35,36].
The organization of the paper is as follows : In section 2, I derive the photon wave function by quantizing the classical energy equation of special relativity, and then refute all objections to a photon wave function. The momentum operator is found invertible and the eigenvalue of this inverse operator is just the reciprocal of
eigenvalue of the operator. In section 3, I derive another wave function, but this time it is
motivated by a purely mathematical reasoning : The derivatives d dx φ1 and d dt
φ
2need not
necessarily be constants; they might be functions of x and t respectively. This yields a space- and time-varying speed of light in complex spacetime manifold. The controversy regarding the validity of VSL theories, in the context of a varying dimensionless number α, loses strength once it is realized that real-valued meter rods and real time-ticking clocks are of no use in measuring a complex-valued speed of photon traveling in complex spacetime. The varying speed of light extracted in real spacetime is found greater than the speed of light in vacuum, c 0. This immediately made us consider tachyons --- the dreaded objects with imaginary mass, and not detected to date. But I have made a significant change in their property : Their imaginary energy and momentum are retained. These are not new concepts in Quantum Mechanics
embedded in complex spacetime (see Ref [37] for details). Tachyon mass is kept real. A tachyon cannot be found in complex spacetime because the latter is operationally inaccessible. It cannot be found at a measurement event [37] since it pops up as a photon there. While discussing these I have deduced an explicit expression for the
relativistic mass operator $()($)m
v m v r r =. Next I turned to the problem of finding the least possible invariant mass of a tachyon in section 4.To this end, an ansatz is inserted into Klein-Gordon
equation to obtain a four-component wave function. Arguments have been advanced to take away the sting of all the objections to its use as a valid wave function. The non-zero mass of photon (or, its complex spacetime masquerader, tachyon) has been found to be 1.7×?1038gm which is exceptionally close to values obtained in GRB events. The quantum-mechanical analog of the classical relation E Mc =2 now reads
E Mc M v c
=??L N M O Q
P 2
22281h .
From the varying speed of light I derive the the most important relation concerning variation in fine structure constant, in section 5 :
?ααλπ=?=?F H G I c t l c t p 0022
3
where the look-back time is t and the constant λ was later found to be proportional to Planck length l p . The above equation explicitly shows that a time-varying α occurs at
Planck scale.
Section 6 describes the Quantum Gravity theory of photons derived from quantum potential that introduces slowing down of older photons. The key ingredient in quantum gravity of photons is their non-zero invariant mass revealed by the theory. An important finding is that the Schwarzschild radius for photon is equal to l
p
2.
There is indication in the theory that ′αof string theory is also equal to l
p
2. As a refinement of Heisenberg`s position-momentum uncertainty relation we prove a theorem stating that a particle cannot have a definite position; nor it can have a definite momentum. Complete uncertainty ! The theorem in general states that an incompatible observable cannot have a definite value.This introduces a formidable tool in quantum physics : It immediately removes big bang singularity of Quantum Cosmology, singularities owing to notorious zero-distance interactions in QED, and Schwarzschild black-hole singularity.
I derive tachyon wave function in this section. The half wave number along the imaginary axis drove us to the idea that a single vertical wave is confined in an inaccessible region of Planck length. This oscillation is traveling with a velocity c(x,t) along the real x-axis. This describes an open string whose lowest excitation produces tachyons. I have shown that tachyons in complex space-time show up as photons at the measurement point. This is equivalent to tachyon condensation [38,39] which carries the tachyon to a stable state, viz., photon at the measurement point (MP).
In the same section we discuss the possible Lorentz non-invariance owing to non-zero invariant mass of photon. While the (classical) theory of special relativity does not admit a preferred reference frame, the measurement problem of Quantum Mechanics[40] poses the problem of preferred basis[41] which can be resolved only by stating that the preferred reference frame(the frame containing the device particle at the MP as the origin) is chosen by the observer or measuring
device[37].This is also supported by Copenhagen Interpretation[40]. Quantum Mechanics requires ten-dimensional spacetime manifold [37] to describe a quantum system. Of these, five are imaginary dimensions which are obviously compactified since measurements always take place at real spacetime.The negative norm states of string theory are no longer disastrous as Quantum Mechanics permits negative probability[37]. String theory is therefore on the right track !
Finally, section 6 compares the predictions of our quantum gravity theory of photons with the experimental results gleaned from observational data
of recent years showing the persistent variations ?α
α
and
1
α
αd
dt
. These include the
quaser and atomic clock experiments. The theory is in good agreement with the experimental results. This perhaps fulfils a hope that a quantum gravity theory has been experimentally confirmed. The experimental results have been provided with a consistent theoretical underpinning that stems from Quantum Mechanics.
2. PHOTON WAVE FUNCTION
There are clear-cut objections to a photon wave function [42A,43,44]. I shall discuss them only after deriving the wave function. The energy equation of special theory of relativity is
E p c m c 22224=+
For a photon, the invariant mass m is zero. Hence the quantized form of the above equation is
$$E
p c 222ψψ= (1) Being an operator equation, the above equation cannot be simplified to
$$E
pc ψψ=.
Replacing $E
and $p by their explicit forms, Eq.(1) reads
??=??2222
2ψψt c x
. (2) The concept of wave function of photon is not new, but it has only found reference within the description of second quantization with creation and annihilation operators [42]. For the wave function of photon, we assume the splitting:
ψψψ(,)()()x t x t =12
Since ψ(,)x t is complex, there is no reason to believe that any one of ψ1()x or ψ2()t will be real. In general, both will be complex. Therefore,
ψφ111()()exp[()]x R x i x = ,
where P x x R x 11212
()()()==ψ. Also,
ψφ222()()exp[()]t R t i t =
where P t t R t 22222
()()().==ψ Consequently, the ansatz for photon wave function is
ψφφ(,)()()exp[()()]x t R x R t i x i t =+1212. (3)
The probability density is
P x t P x P t R R (,)()().==121222
(4)
The definitions of wave number κ and ω imply
κφφφκ=??==±x d dx or d dx
11,, (5) and
ωφφφ
ω=
??=
=±t d dt or d dt
22,, (5)
since
φφφ(,)()()
x t x t =+12.
There are two possibilities:
d x dx and d t dt
φφ12()()
may be constants, or, these may be functions of x and t respectively.
We first consider the case when κ and ω are constants. For a photon traveling along +x direction, the obvious choice is
d dx and d dt
φκφ
ω12==?,, (7)
which is just one of four alternative choices. Inserting the ansatz, Eq.(3), in Eq.(2), one obtains (provided ψ≠0)
122222222R d R dt i R dR dt ??=ωω c R d R dx i R dR dx
212
1112
12+?L N M O Q
P κκ
= a complex constant = a ib say 11+,, (8)
where a 1 and b 1 are real. Equate the real parts of both sides of the first and last parts of Eq.(8) to obtain
1222
2
21R d R dt
a =+ω (8a)
which yields
R t A a t 21212
()exp[()]=±+ω (9)
with a real constant A. Equating the imaginary parts of both sides of the first and last parts of Eq.(8), one obtains
222
1ωR dR dt
b =?.
This yields
R t B b t 21
2()exp
=L N M O Q
P ω (10) where B is a real constant. At t = 0, Eqs. (9) and (10) give
R A B 20().==
Equating the right sides of Eqs. (9) and (10), one has
±+=?a b
1212ωω
. (11)
Now equate the real parts of the second equation of Eq.(8) to find
c R
d R dx
a 2121
2
12=+ω. (12)
The solution is
R x B x a c 1
12
2
()exp =′±+L N
M M O Q
P P ω (13)
where ′B is a real constant. Similarly, equating the imaginary parts of both sides of the second equation of Eq.(8), one arrives at
2211
1c R dR dx
b κ= (13a)
which in turn shows that
R x A b x
c
112()exp .=′L N M O Q
P ω (14)
′A is a real constant. At x = 0, Eqs.(13) and (14) gives
′A =
′=B R 10().
When a photon approaches a measurement point (MP) at x, P x 1() will obviously decrease with increasing x [see the probability wave diagrams in Ref.37]
dP dx d dx R R dR
dx
1121120== . Since R x P x dR dx 11 1 0()(),=> will be negative. Eq.(13a) then requires b 10<. Let b b 11=?′, where ′>b 10. From Eqs.(14) and (10) respectively, the amplitudes become R x A b x c 112()exp =′?′L N M O Q P ω (15a) and R t B b t 212()exp =′L N M O Q P ω (15b) The above two equations combine to form the photon probability density for an incoming photon [ Fig.(1)] : P x t R R P (,)(,)==1222 00exp (),?′?L N M O Q P b c x ct 1ω for x ct ≥ (16) When a photon recedes the MP (The point x is called the MP because ψ(,)x t or P x t (,) is measured at this point at a distance x from the origin) P x 1() would increase with increasing x, so dP dx R dR dx 11120=> implies dR dx 10>. Eq.(13a), as also Eq.(10) require b 10>. The result for a receding photon is [Fig.(2)] P x t P b c x ct (,)(,)exp ()=?L N M O Q P 001ω, for x ct ≤. (16a) Eqs. (16) and (16a) show that probability density is a wave traveling with velocity c (equal to photon speed)along +x direction. The dynamics of the probability wave has been described in detail in the context of quantum measurement problem in Ref.[37]. Time is measured such that when the photon is at x = 0 , t = 0.Hence, P(0,0) = 0. Referring to Fig.(1), where the photon is at a distance ct from the origin at time t, we normalize P(x,t) : 1 = P x t dx b c x ct dx ct ct (,)exp ()]=?′ ?L N M O Q P ∞∞z z 1ω. Integration yields the value ′=b c 1ω. From Eq.(16), the probability density is P x t x ct (,)exp (=? for x ct ≥. (17) If position of the MP is measured from the photon [Fig.(1)], then s x ct =?≥()0 and probability of finding the photon at a distance s from it may be found from Eq.(17): P s s ()exp(),=? for s ≥0. (18) It is straightforward to show that P x t x ct (,)exp(),=? for x ct ≤ (19) The fact that a photon is quantum-mechanically a probability wave (as the above equation says) rules out the objections to a photon wave function[42A,43,44]. While a conventional wave function of photon, e.g. Landau-Peierls function[45], consider it as having electromagnetic origin, we here explore only the ontological basis of a photon through its probabilistic origin. While photons are quanta of electromagnetic field, they are also quanta of probability field P(s) ,described by Eq.(18). This equation also supports the observation of I.Biyalinicki-Birula [46] that photo-detection probability falls off exponentially. There are mainly two objections to photon wave function : (i) No position operator exists for photon, (ii) While the position space wave function may be localized near a space-time point, the measurable quantities, like the electromagnetic field vectors, energy and photo- detection probability remain spread out. To refute the objections, I first derive the photon wave function. Eq.(7) implies φκ1(),x x = and φω2()t t =? where I have ignored the arbitrary constants of integration. Making use of these and Eqs.(3),(17) and (19) the photon wave functions in the two domains take the following forms : ψκω(,)exp ()x t x ct i x i t =??+?L N M O Q P 1 2 , for x ct ≥ (20) and ψκω(,)exp ()x t ct x i x i t =??+?L N M O Q P 1 2 , for x ct ≤ (21) At the MP, viz. x = ct, the wave function becomes a stationary wave ψκωx x c i x i t ,exp()F H G I K J =? (22) so in the measurement event at spacetime point (x = ct, t) the probability density is P x x c ,F H G I K J =1, implying that a whole photon has been found at x at time t = x/c. The MP characterized by x = ct is the one-dimensional analog of the three-dimensional MP r x y z ct =++=222. It is interesting to note that measurement event is recorded at null interval : c t x y z 222220???=. The intervals shown in Eqs.(20) and (21) are respectively space-like and time-like. Quantizing the classical relation of position s p of a photon at time t 0 , s ct p =0 one obtains $s ct p ψψ=0. (23) For a photon, c E p =/. Symmetrizing it and then inserting in Eq.(23), I obtain the eigenvalue equation after quantization: $$$$$s Ep p E t p ψψ=+??12 1 10 d i (24) To be valid, the above equation requires the existence of invertibility of $p . To investigate this, we find the norm $p i x i x ψψψ=??????h h = h κψ21 4 + where ψ is described by Eq.(20). If b is a positive number such that h κ21 4 +≥b , then $p b ψψ≥ for every ψ∈dom p ($). Therefore $p admits a continuous inverse $p ?1. To find its eigenvalue, note that $$p p ?=1ψψ, or, using the explicit form of ψ given in Eq.(20), the above equation results in $p i ?=+L N M M M O Q P P P 1 1 12ψκψh h . (25) The required eigenvalue of $p ?1 is 1 12h h κ+L N M M M O Q P P P i . We are now in a position to study Eq.(24).Using Eq.(25), Eq.(24) gives $$$s t E i p i t p ψψκψ=++??F H G I K J L N M M M O Q P P P ?01 212h h h = i t c i i h h h 0 22212?F H G I K J +F H G I K J ωψκ = ct 0 ψ (26) where,as usual, ψ is described by Eq.(20). This eigenvalue equation unambiguously proves that the position operator of a photon exists and its explicit form at time t 0 is $$$s t i t p p i t p =??+??F H G I K J ??0112 h h . (27) while its eigenvalue at that time is simply ct 0.The eigenvalue is real. This sets apart photons from all other particles in Quantum Mechanics. While I have shown elsewhere [37] that all quantum systems travel in complex spacetime except at the measurement event, photons always move in real spacetime.This might provide a reason why photons have the maximum permissible speed in real-dimensional nature! The other objection to a photon wave function may be refuted in quantum-mechanical terms. It is reiterated that the photon wave function derived here has no relation whatsoever with the wave function derived from electromagnetic inputs. Here, the photon is a probability wave ------ a generic mathematical wave in probability space. Indeed all quantum systems are probability waves [37] when left to evolve i.e. when they are not subjected to measurements. The probability fields of different quantum particles do not interact among themselves because these are mathematical (probability) fields. The observables of a photon may be calculated using the appropriate operators on the photon wave functions, Eqs.(20) and (21). Of course, the corresponding eigenvalues (except position) may be complex. This only shows that the corresponding operators are non-Hermitian normal operators which transform into Hermitian ones at the MP [37]. At the MP the measurement device will record real numbers ---- so there is no problem. Let us see that it is really so. In general , when the photon is not at the MP, i.e., x ct ≠, $E i t i c ψψωψ=??=+F H G I K J h h h 12. But at the MP, x = ct, and ψκω=?exp i x i t b g , and so $E ψωψ=h . This reveals the transformation of non-Hermitian operators into Hermitian ones at the MP. The complex eigenenergies need not be cofused with those characteristic of decay or growing Gamow vectors. These are mere consequences of the complex spacetime manifold in which the quantum particles live. So far I have discussed the case when d dx φ1 and d dt φ 2are constants. Now I consider the fact that these derivatives need not mean that they are not functions of x and t respectively. This possibility introduces tachyons in complex spacetime. 3. WHEN d x dx φ1 b g AND d t dt φ 2 b g ARE NOT CONSTANTS I therefore rewrite the wave number and frequency as d x dx φ1b g = ±κx b g , and d t dt φ2b g = ±ωt b g and for a photon traveling in +x direction, I choose d x dx φ1b g = κx b g and d t dt φ2b g = ?ωt b g Since ωκt x c x t c x c t b g b g b g b g ==(,)12, where separation of variables have been assumed, one finds κx b g = 1 1c x b g and ωt b g = c t 2b g . Keeping these in mind, I insert Eq.(3) in Eq.(2) and find 122222 222 R d R dt i R dR dt i d dt ???ωωω = c 2121212112R d R dx i R dR dx i d dx ++?L N M O Q P κκκ (28) The above equation generates four equations when real and imaginary parts of both sides are equated : 11222222 2c R d R dt ?L N M O Q P =ω c R d R dx 1212 1 221?L N M O Q P =κ a real constant = a , (29a) ?L N M O Q P +L N M O Q P =12222c d dt R dR dt ωω c d dx R dR dx 1211 2κκ+L N M O Q P = a real constant = b . (29b) Substituting [/]1κx b g and ωt b g for c x 1b g and c t 2b g respectively the following equations are obtained : 1222 2 22R d R dt a ?=ωω (30a) 11212 22 R d R dx a ?=κκ (30b) d dt R dR dt b ωωω+=?222 2 (30c) d dx R dR dx b κκκ+=211 2 (30d) From Eq.(30c) , a little rearrangement gives d dR R b dt ωω ω+=?22 2 which after integration results into R t A b dt 22 b g =?L N M O Q P z ωωexp (30) where A is a real constant. Eq.(30d) is similar to Eq.(30c) except for the sign on the right side. Hence its solution may be immediately written down : R x B b dx 12 b g =L N M O Q P z κκexp (31a) φκω12x t dx dt b g b +=?z z (), Eqs.(30) and (31a) help write down the photon wave function from Eq.(3) : ψωκκωκωx t AB b dx b dt i dx i dt ,exp b g =?+?L N M O Q P z z z z 22 (31b) and P x t D dx dt ,b g e = L N M O Q P ?z z ωκκω (31b) where AB = D and x c x t dt ≥z ,.b g From Eq.(31b), P(0,0) = 1 means D =ωκωκ000 b g b g =. Hence P x t dx (,)(= ?z z ωκωκ κω00 But κωdx dt ?z z =κ((,)x c x t ?z and x c x t dt ≥z (,) implies that c x t ?≥(,)0, so, P x t b dx cdt (,)exp = ?ωκωκ κ00 l q . (32) κ()x is positive as the photon is traveling along the +x direction. Now a look at a similar photon wave function of constant speed, viz. Eq.(20) for x ct ≥ suggests that b in the exponent of Eq. (32) must be negative in order to check the diverging P(x ,t). We set b b =?′, where ′b is positive. Eq. (32) then describes probability density as an inhomogeneous wave traveling with a varying speed c(x ,t ) along +x direction. The photon wave function may now be written down from Eq. (31b): ψωκωκ κω(,)exp x t b c c dt c b dx c i dx i dt =′?′+?L N M M O Q P P z z z z 00 121 1 22 (32a) or, ψωκωκ κωκω(,)exp x t b dx dt i dx i dt = ? ′ ?+?L N M O Q P z z z z 002o t (32b) where x c x t dt ≥z (,). The rule for finding the value R of an observable $R in a particular measurement is [37] : R = $,R ψψ and therefore, the photon energy in a particular measurement , calculated from Eq. (32b), is E E t i b t i d dt r ==+′?$()()ψψ ωωωω h h h 122 (33) The photon momentum may be similarly found : p p x i b x i d dx r = =+′+$()()ψψκκκκ h h h 122 (34) The energy and momentum values of a photon are related as E p c x t p c x c t r r r ==(,)()()12 (34A) Inserting the values of E r and p r from Eqs.(33) and (34) respectively, in Eq.(34A) and replacing ω()t and κ()x by c t 2() and 1 1c x () one arrives at h h h h h h ωωωωκκκκ ()()()()t i b t i d dt c c x i b x i d dx +′?=+′+L N M O Q P 12212212 (34B) which is separated into real and imaginary parts and a little rearrangement leads to the important result : 12221c dc dt dc dx ==λ (34a) where λ is a real constant. The above two equations yield the following solutions : c x c x 110()=+b g λ (35) c t c tc 2220 10 ()=?b g b g λ (36) The space-time varying speed becomes c x t (,)=2 (36A) From Eq. (34a) one finds ??==c t c dc dt c c 12122 λ c c x c c dc dx c c ??==122112 2 λ, and the above two equations easily presents a differential equation for varying speed of light : ??=??c t c c x (36a) Varying speed of light (VSL) theories [47,48,49,50,51] have always received fatal blows while trying to interpret the varying fine structure constant results. While interpreting the varying fine structure constant results the lethal line is that all lengths, masses and times are measured in dimensionless numbers (multiples) of unit length, mass and time. A change in a dimensionful quantity cannot rule out the possibility that the meter rods ,clocks etc have undergone similar changes in lengths ticks etc. No experiment can distinguish ! Therefore change in a dimensionful quantity does not point towards a change in a dimensionless quantity like α. But all these arguments hold only when we measure something like speed of light with `meter` of a meter ruler and `second` of a clock, which are all real numbers.That is , the measurements involve four-dimensional real spacetime. But here, we find that something extraordinary happens. From Eq. (34A), the varying speed of light is c x t E p t i b t i d dt x ib x i d dx r r (,)()()()()== +′?+′+ h h h h ωωωω κκκκ122122 (36b) where we have used Eqs.(33) and (34). This speed c(x ,t ) is a compex quantity, and so it cannot be measured by real meters and seconds. In fact these speed is not measurable. Whenever we measure speed of light this space-time varying complex speed transforms into a real valued constant speed c c (,)000=, which is the present speed of light in vacuum. To prove how this measured value of a complex c(x ,t ) transforms into a real c(0,0) we recall that the criterion of measurement is spelt out as x c x t dt =z (,) (36c) from which we find x c x c t dt t 120 ()()=z = a real constant (36d) Differential of the second equation gives c t dt 20()=. Since c t 2() cannot be zero (otherwise c(x ,t ) would be zero), dt =0, i.e., t t =0= a constant at the measurement event. Since dt = 0, differentiating Eq.(36c) yields dx c x t dt ==(,)0, i.e., x = x 0 = a constant. If we want to measure c(x ,t ) at the MP, then we replace x and t in Eq. (36b) by x 0 and t 0. Now we use the relation ωκ()()(,)t x c x t 0000= in Eq. (36b) to obtain c x t (,)= c x t (,)00 (36e) If we set the present time as t = 0 = t 0, then Eq.(36d) gives x = 0. Hence from Eq.(36e) the speed of light at the MP is c c (,)000=. . Gravitational redshift of light rising away from a distant quaser is caused by a decrease in frequency : d dt ω < 0, i.e., dc dt 2 < 0. Eq.(34a) now says λλ=?<00. If we assume the present spacetime as origin (t = 0, x = 0) located in the earth-bound laboratory, then the look-back time t =?<Τ0, and the look-back distance of the quaser,(- x) constitutes the measurement point. Eqs.(35) and (36) change, but are not altered in form c x c x 1100()(),=+λ c T c c T 2202010()() ()=?λ From above, c x c 110()()> and c T c 220()()> , whence c x T c x c T c c c (,)()()()()=>=1212000 The varying speed of light is then c x T c x c c T (,)()() ()=+?L N M O Q P 102020010λλl q Since the above equation and Eq.(36A) are similar in all respects we shall drop λ0 and T in future discussion, and instead continue to use λ and t and Eq.(36A). I have thus proved that c ( x t ) > c 0, an astonishing result that says, older photons had higher speed than c 0. Special theory of relativity categorically says those older particles were certainly not photons. But tachyons [52,53] may be suitable candidates satisfying this faster-than-light speed. Note that c x t c c c x t c E t (,)()()() ===1211ωh and this implies an energy-dependence of speed of light; quantum gravity models suggest just this [54,55]. A few difficulties are now to be resolved once I propose tachyons[56]. An infinite amount of energy is required to slow a tachyon down to the speed of light. But theory requiring this is the classical special relativity. In Quantum theory I have proved through Eqs. (36c),(36d) and (36e) how c ( x t ) , a higher speed, transforms into the lower speed c 0 at the MP. The violation of causality by tachyons never takes place because emission of a tachyon at a measurement event E and its absorption at event E’ actually involve photons, not tachyons. This is due to the fact that tachyons show up as photons at the measurement events. This is possible because the least possible speed of tachyon is speed of light. The events E and E’ involve photons ,so causality is not flouted. As tachyons move in complex spacetime they will forever remain unobservable. The 26-dimensional string theory requires tachyons.The pernicious negative norm states are however no longer dreaded objects as the latter have been shown to be perfectly compatible with Quantum Mechanics [37]. Superstring theory has replaced bosonic string theory because the latter suffered from tachyonic instability. But it will become clear later that tachyons in complex spacetime having real (and not imaginary) mass, with, of course, imaginary energy and momentum (not surprising in complex manifold) may not be a disaster. For tachyons with v > c , energy and momentum p imv v c =?2 21, (37A) E imc v c =?2 2 2 1 (37B) are imaginary quantities.This is not anarchy ---- as tachyons move exclusively in complex spacetime and never show up at real spacetime. I do not attribute the peculiar tachyon properties in the conventional way: assigning imaginary invariant mass to tachyons. Rather I keep tachyon mass real while E and p are imaginary even in the classical domain, as tachyons have never been detected at classical spacetime points at a measurement event. Quantizing Eq.(37B) $$$E i t imc v c im c r ψψψψ=??=?=h 2 2221 (37) where $v is assumed a multiplicative operator (as it works!) and $m r is the relativistic mass operator. Explicit forms of two operators are in order : From Eq.(37), the operator $$γ=?=??1122 2v c mc t h (38) which may be of immense help in quantizing important classical relations of special relativity. From the same equation , the relativistic mass operator is found: $$m m v c c t r =?=? ?22 21h (39) To find the tachyon wave function, I insert the ansatz 第37卷第9期2008年9月 光 子 学 报 ACTA P HO TON ICA SIN ICA Vol.37No.9 September 2008 Tel :02928220149828313 Email :warltszhang @https://www.doczj.com/doc/0e1097090.html, 收稿日期:2007204228 一维光子晶体带隙结构研究 张玲,梁良,张琳丽,周超 (西安建筑科技大学物理系,西安710055) 摘 要:在考虑介质色散的基础上,研究了介质层厚度对光子晶体带隙结构的影响.利用传输矩阵法,计算了以Li F 和Si 两种材料组成的一维光子晶体带隙结构.结果表明,介质层厚度的增加会引起禁带的红移,厚度减小会引起蓝移.分析了含空气缺陷层、金属缺陷层的光子晶体结构,发现空气缺陷层对带隙结构的高反射区域变化不大,而在低反射区域,反射系数为零的波带之间出现了两边反射系数增加,中间反射系数减小的情况.在金属缺陷层的带隙结构中,金属对整个波长范围光的吸收作用不同,金属对低反射区1.6μm 、1.85μm 处透射率较大的透射光吸收作用明显,而在1.28～1.38μm 处透射率波长区间,几乎无吸收. 关键词:光子晶体;色散;带隙结构;空气缺陷层;金属缺陷层中图分类号:O734 文献标识码:A 文章编号:100424213(2008)092181524 0 引言 微加工技术的进步,使得光子晶体[1]在理论和实验研究上取得了重大进展,利用光子晶体可以制造出光通信中的许多器件,如光纤、微谐振腔,品质优良的光子晶体滤波器、集成光路等等[223].实验室一般采用不同折射率介质在空间的周期性排列形成光子晶体,Ward 等人提出一种增强块状金属反射能力的方法,他们预测含有Al/玻璃层的一维金属/电介质光子晶体比块状Al 的反射能力更强[4].对Au/MgF 2光子晶体透射性质的研究发现,周期性结构产生的透射共振使得光通过金属层的透射率大大增强,并有效抑制了吸收.通过控制金属层和电介质的厚度以及周期数,可以调节透射区域的波长范围、宽度和陡度[5].如果在光子晶体中引入缺陷,可使光子局域化[6],在有缺陷层的一维光子晶体(AB )n D m (BA )n 的带隙结构发现随着缺陷层厚度的增加,在禁带中出现的缺陷模向低频方向移动[7].还有一些金属/电介质光子晶体可以对某些晶体的闪烁光谱进行修饰,使得其对慢衰减成分的相对抑制比大大提升等等[8].本文在考虑色散关系的基础上对于LiF 与Si 构成的2元一维光子晶体的带隙结构进行了研究,通过改变介质层的厚度,分析了其带隙结构的变化,另外当该结构的光子晶体中有空气缺陷层、金属缺陷层时,其带隙结构的变化[2],并对计算结果做了分析. 1 理论模型 典型的光子晶体是由两种不同介电常量(εa ,εb ),厚度为(d a ,d b )的材料交替排列的其结构如图1,根据光在介质薄膜传播的传输矩阵方法,在第一 介质中的传输矩阵为 M a = cos δa isin δa /ηa i ηa sin δa cos δa (1) 图1 一维光子晶体模型 Fig.1 The structure of 12D photonic crystal 在第二介质中的传输矩阵为 M b = cos δb isin δb /ηb i ηb sin δb co s δb (2) 式(1)、(2)中δj =2πn j d j cos θ/λ,n j 、d j 、θj ,分别为第 j 层(j =(a ,b ))的折射率,介质层厚度,入射角, λ为真空中的波长,对于TE 波:ηj =n j cos θj ,对于TM 波ηj =n j /co s θj , 对于整个光子晶体的传输矩阵,若取层的对数为n ,则 M =(M a ,M b )n = M 11M 12M 21 M 22 (3) 设光子晶体周围材料的折射率为n 0,对于TE 波η0=n 0co s θ0,光在光子晶体传播时的反射系数和透射系数分别为 r = (M 11+M 12η0)η0-(M 21+M 22η0)(M 11+M 12η0)η0+(M 21+M 22η0) (4) 科研文献调研报告 题目：一维光子晶体的能带结构研究 学院：＿＿理学院＿ 专业：＿＿光信息科学与技术＿＿ 班级：＿2008级 学号：＿ 080701110083 学生姓名：＿＿李辉＿＿＿＿＿指导教师：＿＿徐渟＿＿＿＿＿ 2012年3月14日 一维光子晶体的能带结构研究 摘要： “光子晶体"的概念是1987年S．John和E．Yabloncvitch分别提出来的。而在当今世界，科学家们在不断研究电子控制的同时发现由于电子的特性，半导体器件的集成快到了极限，而光子有着电子所没有的优越特性：传输速度快，没有相互作用。所以科学家们希望能得到新的材料，可以像控制半导体中的电子一样，自由地控制光子。与此同时随着科学技术的发展特别是制造工艺技术的发展，使得光子晶体的制造不仅变得可能，还得到了长足的进步，在可见光及红外波段可以制成具有所需能带结构的光子晶体，实现对光的控制。因此近年来光子晶体得到深入广泛的研究与应用。 关键字：光子晶体能带结构半导体器件 The Investigation on the Band Structures of one-dimensional photonic crystal Abstract： The concept of"Photonic crystals" was put forward byS.John and E.Yabloncvitch in 1987.But nowScientists constantly study electronic control and find that the integration of semiconductor devices has been the limit because of the characteristics of the electronic.And the photon has the advantage of high speed,no interaction, which electron does not have.So scientists want to get 闽江学院 本科毕业论文(设计) 题目一维光子晶体的禁带宽度分析 学生姓名 学号 系别电子系 年级03 专业电子科学与技术 指导教师 职称副教授 完成日期2007.05.16 目录 摘要 (2) ABSTRACT (3) 第一章绪论 (4) 1.1什么是光子晶体？ (4) 1.2光子晶体理论计算方法 (5) 1.3光子晶体的应用 (8) 第二章一维光子晶体基本理论 (9) 2.1光子禁带的产生 (9) 2.2一维光子晶体的特征矩阵 (11) 第三章一维光子晶体带隙变化规律的研究 (13) 3.1带隙随厚度比的变化 (13) 3.2带隙随折射率差的变化 (16) 3.3带隙随角度的变化 (19) 3.4厚度比与折射率差同时变化下的最大带隙 (22) 总结 (24) 参考文献 (25) 摘要 光子晶体的研究领域非常广泛，涉及到光学的方方面面。由于它所具有的特殊的性质，故被称为光的半导体，足见它对光学领域的影响力。虽然这个领域的工作也才刚开始10年多一点，但是进展非常地快。通过对这个领域的深入研究.不仅对光子晶体研究本身有意义，而且对光学领域的理论发展也具有重要的价值。使得人们对光的理解更加深入。 介绍了一维光子晶体的基本概念和原理系统综述了对一维光子晶体的研究进展和应用前景。 作为一维光子晶体的应用基础,一维光子晶体的禁带是研究的重点。一维光子晶体的带隙决定了工作频率范围，因此研究其带隙变化规律是其应用的关键，通过改变各种参数确定带隙的依赖因素及其定量关系。 通过传输矩阵的方法分析了一维光子晶体禁带的特性,讨论了影响带宽的因素，说明了相对带宽对光子晶体设计的重要性。在这个基础上讨论了扩展一维光子晶体带宽的方法,：1、使各层介质的厚度d微微变化，形成规则递增，达到展宽禁带的目的。2、角度 逐渐变化，使晶体在角度域化互相叠加，达到扩展带宽的目的。3、使晶体的折射率n1逐渐变化（n2＝4.6），达到扩展带宽的目的。通过画出改变各种参数的情况下的带隙曲线图，得到带隙随各参数变化的规律，从而达到对一维光子晶体带隙变化规律的分析。 关键词:光子晶体;光子禁带;相对带宽;展宽。 一维光子晶体层状碘化铅/碘甲基氨的色散关系 自1987 年Yablono vitch[ 1 ] 在周期性排列的电介质中发现光子禁带以来, 人们对光子晶体这种人工结构已做了大量的研究工作。一维光子晶体, 其结构简单（图示1）, 易于制备, 可以设计滤波器、薄膜太阳能电池等光电子学器件的常用结构。 使用CVD法制备卤化铅（碘化铅）层状结构，后期退火在每层碘化铅中加入碘甲基氨，由于二者的介电常数相差较大且呈周期排布所以在堆垛方向上形成一维光子晶体（图示2）。 通常描述光子晶体能带结构的物理参量主要是透射谱、反射谱及其）（k ω色散关系.本文中我们用平面波展开发计算色散关系[2]. 光子晶体理论分析中应用最早、最广的一种方法就是平面波展开法。在计算光子晶体光子能带结构时,平面波展开法直接应用了结构的周期性,将麦克斯韦方程从实空间变换到离散傅立叶空间,将色散关系计算简化为对代数本征值问题的求解. 假设光子晶体处在无源空间, 且是由各向同性、无损耗、非磁性、无色散的线性介质组成 入射波t i e x E t x E ω-=)(),( 由麦克斯韦方程给出其波动方程 2222),()(,t t x E a x x t x E ??=??ε）（ 图2 碘化铅层状结构SEM 图 削去时间 )()(-2222x E c x x x E εω=??）（ a 为晶格常数，)(x ε为周期性介电函数， nm a nm a a a a 4040212 1==+= ???<<<<=a x a a x x 1211 ,0,)(εεε 1a 为碘化铅厚度，2a 为碘甲基氨厚度，假设二者相等，根据图2可估算大概尺度为40nm 1ε为碘化铅介电常数，2ε为碘甲基氨介电常数，查阅资料取31=ε62=ε 将周期函数)(x ε做周期展开 ∑∞-∞== n x a n i n e x πεε2)( 其中 ?-=a x a n i n e x a 02)(1πεε 积分得 ???????≠??????--=+=-0,1)(20,12212211n e n i n a a a a a a n i n πεεπεεε 将E(x)展开得到布洛赫波的形式 ∑∞-∞=+= m x a m k i e m B x E )2()()(π 将②③带入①中 ① ② ③ 一维光子晶体的研究方法----传输矩阵法 1：绪论 1．1：光子晶体研究的意义 在以前对半导体材料的研究导致一场轰轰烈烈的电子工业革命，我们的科技水平有了突飞猛进的发展，并为此进入了计算机和信息为标准的信息时代。在过去的几十年里，半导体技术正向高速，高集成化方向发展。但这也引发了一系列的问题，比如电路中能量损失过大，导致集成体发热。此外，由于高速处理对信号器件中的延迟提出更高的要求，半导体器件的能力已经基本达到了极限，为此科学家们把目光从电子转向广光子。这是因为光子有着电子所不具备的优势：1.极高的信息容量和效率。2.极快的响应速度。3.极强的互连能力和并行能力。4.极大地存储能力。5.光子间的相互作用很弱，可极大地降级能力损失。但是与集成电路相比，科学家们设想能像集成电路一样制造出集成光路，在集成光路中，光子在其中起着电子的作用，全光通过。光子计算机将成为未来的光子产业，集成光路类似于电子产业中半导体的作用，光子产业中也存在着向集成电路的器件一样的集成光路——光子晶体，光子晶体的研究不仅仅是光通讯领域内的问题，同时也对其他相关产业将产生巨大的影响。 1.2：光子晶体的概念及应用 光子晶体是八十年代未提出的新概念和新材料，迄今取得了较快的发展，光子晶体不仅具有理论价值，更具有非常广阔的应用前景，这个领域已经成为国际学术界的研究热点。 控制光子是人们长期以来的梦想，光子晶体能帮助人们实现这一梦想。1987年Yablonol itch在讨论如何控制自发辐射和John 在讨论光子局域化时各自独立的提出了光子晶体的概念。他们所讨论问题的共同实质是周期性电介质材料中光传播的特性，根据固体电子能带理论，晶体内部原子呈周期性排列，库仑场的叠加产生周期性势场，当电子在其中运动时受到周期性势场的布格拉散射而形成的能带结构，带与带之间有带隙，称为禁带。能量落在禁带中的电子波不能传播。与此相仿，当电磁波在周期性电介质结构材料中传播时由于受到调制而形成能带结构——光子能带结构，其带隙称为光带隙（PBG:photonic band gap）。此具有 龙源期刊网 https://www.doczj.com/doc/0e1097090.html, 一维光子晶体的应用发展 作者：江帅璋 来源：《新教育时代·学生版》2016年第33期 摘要：一维光子晶体是介质特定的在一个方向上具有周期性的结构，在另外的两个方向上却是均匀性分布的。结构比较简单的一维光子晶体一般是两种介质交替叠层而形成的，这种一维光子晶体在垂直于介质层平面方向的介电常数是随空间位置的改变而改变的，而在平行于介质层平面方向的介电常数并不随空间位置的改变而改变。这种光子晶体在光纤和半导体激光器上已经得到了运用，布拉格光纤和半导体激光器的分布反馈式谐振腔事实上就是一维光子晶体。因为一维光子晶体制作简单，结构简单，所以一维光子晶体被大家广泛的关注。在最早期的时候，因为一维光子晶体特定的在一个方向上表现有周期性的结构，所以光子禁带也只在这个方向上出现，之后Joannopoulos和他的同事们根据理论和仿真得到一维光子晶体应该有全方向的三维带隙结构，因此一维光子晶体也能够具备二，三维光子晶体所具有的特性，所以一维光子晶体被人们更加普遍的应用到了研究中。 关键词：一维光子晶体周期性介电常数 一、一维光子晶体的研究进展与应用 一维光子晶体具有制作简易和控制光的传播形式优异性等优势，让一维光子晶体在不一样的研究中得到了广泛的关注。这些年一维光子晶体在研究领域取得了一些明显的进展。因为一维光子晶体拥有三维材料的全向能隙结构，所以可以将一维光子晶体应用到二维和三维器件的设计当中；一维光子晶体有高增益的局域广场以及光延迟效应，能够导致一些非线性效应，比如说谐波的产生、光学双稳态等；并且一维光子晶体也具有超折射现象，而且因为它有控制光模式以及光传输的优异性能，所以一维光子晶体在光子晶体的应用中占据着主要地位。下面我们从三个方面介绍一维光子晶体的特点和应用，分为物理机制和效应两个角度。[1] 1.全向能隙结构 1998年，因为一维光子晶体的边界是有限制的，所以出现了跟二维光子晶体和三维光子 晶体相像的全向能隙结构。虽然金属材料的反射镜的反射率跟入射角度没有关系，但是金属材料是吸收电磁波的，所以金属材料的反射率并不高。以前的多层高反膜会因为入射角度的增加其反射率降低。一维光子晶体可以产生一个不跟入射光偏正方向以及入射角有关联的较宽的全向带隙，解决了金属材料反射率不高的难题。除了反射镜外，一维光子晶体能够普遍的运用到微波天线、透射光栅、光波导等器件的研制中。[2～6] 2.布儒斯特角的控制一维光子晶体带隙结构研究_张玲
一维光子晶体的能带结构研究开题报告
一维光子晶体的禁带宽度分析
固体物理小论文一维光子晶体
一位光子晶体的计算
一维光子晶体的应用发展
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