Download Quantum Mechanics from Periodic Dynamics: the bosonic case

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Donatello Dolce
Quantum Mechanics from Periodic
Dynamics: the bosonic case
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Abstract Enforcing the periodicity hypothesis of the semiclassical formulation of Quantum Mechanics we show the possibility for a new scenario where
Special Relativity and Quantum Mechanics are unified in a deterministic field
theory[1]. From this follows a novel interpretation of AdS/CFT.1
1 Periodic Dynamics
In the old, semiclassical approach to QM, free bosonic waves are supposed to
have angular frequencies ω̄; the energies of the related quanta Ē = ~ω̄ = h/Tt
depend on the inverse of the time periods Tt = 2πRt . To solve a relativistic
differential system (ds2 = c2 dτ 2 = c2 dt2 − dx2 ) we must impose Boundary
Conditions such to minimize the action on the boundaries. For this reason, in
ordinary QFT one assumes fixed values of the fields on the time boundaries,
let’s say at t and t + Tt . For free bosonic fields, we note that it is possible
to satisfy the variational principle, and thus all the relativistic symmetries
of the action, by imposing Periodic BCs (PBCs) Φ(x, t) ≡ Φ(x, t + 2πRt ) [3].
Since in this case the whole information is contained in a single period, we
say that time is compactified on a circle t ∈ S1Rt with radius Rt .
Thus, imposing PBCs as a constraint, the resulting Φ(x, t) is decomposed
as a tower of normal modes with quantized frequencies ωn = nω̄ = n/Rt .
In analogy with the Matsubara and KK theory, it is naturally interpreted as
a tower of energy eigenmodes with energies proportional to the frequencies
En = n~ω̄ = n~/Rt . Since we assume on-shell fields, in the massless case
(ds2 = 0) there is an induced periodicity λx = 2πcRt on the modulo of the
spatial dimensions (c2 dt2 = dx2 ) and thus a quantization of the modulo of the
Institut für Physik, Johannes-Gutenberg Universität, D-55099 Mainz, Germany
E-mail: [email protected]
MZ-TH/09-23
1
This letter (readable with [2]) summarizes results of [1], where details of the
demonstrations, further relevant issues and physical interpretations are reported.
2
momentum as well |pn | = ~|kn | = n|p̄| = n~/Rt c. For dimensional reasons
the proportionality constant is the reduced Planck constant ~ and thus we
naturally get the de Broglie relation Rt (p̄) ≡ ~/Ē(p̄) = 1/ω̄(p̄). Since the
period is related to the inverse of the fundamental energy Ē, and not to the
inverse of an invariant mass as in the KK theory, the compactification must
be regarded as dynamical. The periodicities are described by the four-vector
Rµ = (Rt c, Rx ) and the fundamental mode (n = 1) has four-momentum p̄µ =
(Ē/c, p̄) ≡ ~/Rµ . For this massless field ω̄(p̄) = |p̄|c/~. Thus every mode of
the energy tower has a massless dispersion relation. At small fundamental
momentum |p̄| → 0 the compactification radius tends to infinity Rt → ∞,
i.e. the PBCs can be neglected and the continuos energy spectrum Ē → 0 is
restored. Indeed this is the usual relativistic field which can also be obtained
by putting ~ → 0. On the other hand, in analogy with the relativistic Doppler
effect, increasing the momentum |p̄| → ∞ through interaction or changing
the reference system, one obtains a Lorentz deformation of all the spacetime intervals and thus of the compactification radiuses as well. Since now
Rt → 0, the PBCs are important and we get a well quantized energy spectrum
Ē → ∞ at high frequencies. Actually, this avoids the UV catastrophe in the
blackbody radiation. This limit can also be obtained by putting ~ → ∞ and
therefore can be addressed as the quantum limit of the massless case.
Concerning massive fields (ds2 > 0), it is possible to choose a reference
frame where the proper time is equal to the real time: dτ 2 = dt2 . This means
that also the proper time has induced periodicity Tτ = Tt (0) = h/M̄ , where
we define M̄ to be the mass. By treating the proper time τ = s/c as a virtual
extradimension (XD) for a 5D field with zero 5D mass (dS 2 = c2 dt2 − dx2 −
ds2 ≡ 0), we obtain again a regular energy tower En (p̄) = n~ω̄(p̄) but
now every quantized level has the dispersion relation of a massive relativistic
particle ω̄(p̄) = (p̄2 c2 + M̄ 2 c4 )1/2 /~, just as in the normal ordered second
quantization [4,5] and in perfect agreement with Lorentz transformations.
We define covariant notations: p̄µ ≡ ~/Rµ and M̄ 2 c4 = ~2 /Rτ2 = p̄µ ~/Rµ =
(~/Rµ )(~/Rµ ). In this way we see that the time periodicity Tt (p̄) has an
upper limit Tτ in the rest frame proportional to the Compton wavelength
λs ≡ Tτ c = h/M̄ c, which indeed fixes the mass of the field. Even for a light
boson with the rest mass of an electron, this periodicity is already extremely
fast, 10−20 s, if compared for instance with the characteristic period of the Cs133 atom which by definition is of the order of 10−10 s. Remarkably, evidences
of this intrinsic periodicity of massive particles conjectured by de Broglie [6]
have been indirectly observed in a recent experiment for electrons [7].2
We point out that this theory respects relativistic causality. In fact, the
compactification radius Rt (p̄) is not static but local and dynamical as the
energy: Rt (p̄) ≡ ~/Ē(p̄). Since the given theory is build upon relativistic
waves, we can solve the Green function as usual. Therefore, by turning on a
source in a point, a retarded variation of the energy in a different point is
induced and, just by energy conservation, the field in that interaction point
passes from a periodic regime to another periodic regime. In this way it is
possible to give a chronological order to events and the resulting notion of
time formally fulfils the requirements of SR.
2
Here fermions can be thought of as fields with antiperiodicities Tt (p̄) = h/Ē(p̄).
3
2 Quantum Mechanics
We now study the mechanics of these periodic fields, for simplicity we assume
only one spatial dimension. The first thing we note is that these fields are a
sum of on-shell standing waves with Fourier coefficients An
X
X
X
Φ(x, t) =
An φn (x, t) =
An φn (x)un (t) =
An e−i(ωn t−kn x) . (1)
n
n
n
Actually, we have the typical case where a Hilbert space can be defined.
In fact, the energy eigenmodes φn (x) form a complete set with respect to
Rλ
the inner product hφ|χiH ≡ 0 x dxφ∗ (x)χ(x)/λx . Furthermore, the bulk
EoMs (∂t2 + ωn2 )un (t) = 0, that describe the evolutions along the compact time, can be reduced to first order differential equations which are
nothing else than the Schrödinger equation i~∂t φn (x,√t) = En φn (x, t). Formally, from the Hilbert eigenstates hx|φn i ≡ φn (x)/ λx we can build the
Hamiltonian operator Ĥ |φn i ≡ ~ωn |φn i and thus a time evolution operator
Q −1
′
U (t′ ; t) = exp[−iĤ(t − t′ )/~] which is Markovian U (t′′ ; t′ ) = N
m=0 U (t +
′
′′
′
tm+1 ; t + tm − ǫ), where ǫ is s.t. N ǫ = t − t [8]. Finally, without any
further assumption than periodicity, we plug the completeness relation in
between the elementary time evolutions of the Markovian operator obtaining
a path integral (PI) which is formally the Feynman PI for a time independent
Hamiltonian
! N −1
Z λx NY
−1
Y
i
′′ ′′
′ ′
U (x , t ; x , t ) = lim
dxm
hφ| e− ~ (Ĥ∆ǫm −p̂∆xm ) |φi . (2)
N →∞
0
m=1
m=0
Here ∆xm = xm+1 − xm and ∆ǫm = tm+1 − tm . In fact, also in the usual
Feynman PI the elementary evolutions are supposed to be on-shell [9]
i
U (xm+1 , tm+1 ; xm , tm ) = λx hφ| e− ~ (Ĥ∆ǫm −p̂∆xm ) |φi =
X i
X
e− ~ (Enm ∆ǫm −pnm ∆xm ) = 2π
=
δ ω̄(p̄)∆ǫm − k̄∆xm + 2πn′ . (3)
nm
n′
P
To P
write the last expression we have used the Poisson summation n e−inα =
2π n′ δ(α + 2πn′ ), so that the PI eq.(2) - or similarly the periodic field
itself eq.(1) - can be explicitly
written as a sum over periodicon-shell paths
P
U (x′′ , t′′ ; x′ , t′ ) = 2π n′ δ ω̄(p̄)(t′′ − t′ ) − k̄(x′′ − x′ ) + 2πn′ . Because the
initial and final points are now defined modulo space-time periods, they are
paths with different winding numbers. These on-shell Marcovian and periodic
paths can be ideally cut, translated by periods and combined in such a way
to form paths with the same initial and final points, obtaining the analogous
of the variations around the classical path of the usual Feynman formulation.
However, in the original Feynman PI there is a single classical path linking
the initial and final point, so that the variational principle must be relaxed
to have interference. In our case the field can self-interfere because the PBCs
and the variational principle is preserved since the sum over variations can be
reduced to a sum over periodic on-shell paths. Nevertheless, as in the usual
4
QM, we obtain that the square modulo of the periodic field |Φ(x, t)|2 has a
maximum along the relativistic free particle path where the periodic on-shell
paths have constructive interference.
Taking the non relativistic limit |p̄| ≪ M̄ c, the field can be thought of as
at low intensity so that only the first level is largely populated. Thus (nm ≡ 1;
∀m = 0, . . . , N − 1) we find that the sum over periodic paths eq.(2) reduces
to the usual non-relativistic free particle FPI (plus the de Broglie internal
periodicity). This is a Dirac delta distribution along the particle path. Hence
we have obtained a consistent interpretation of the wave/particle duality and
of the double slit experiment.
The uncertainty relation can be easily obtained (we have standing waves)
by absorbing the periodic invariance of the phase factor Ēt/~ + 2π either as
a time or as an energy variation: ∆E × ∆t = (2π~)2 /Ēt ≥ h, where t ≤ Tt .
This is a direct consequence of the periodic conditions En Rt = n~ which can
be written in a Bohr-Sommerfeld form: the allowed solutions are those with
an integer number of cycles. Following this approach it is straightforward to
obtain the usual solution3 of the quantum harmonic oscillator (QHO) as well
as many others non-relativistic Schrödinger problems. Moreover, generalising
the symmetry breaking mechanism by BCs [3] to a periodic EM field at low
temperature [10], we have an effective quantization of the magnetic flux [11]
which reproduces collective quantum phenomena such as superconductivity.
A further analogy with canonical QM emerges by noting that, from the
given definition of Hilbert space, the expectation value of O(x) is
Z λx
dx X ∗
χ (tf , xf )e−ikχm x O(x)eikφn x φn (ti , xi ) . (4)
hχf |O(x)|φi i ≡
λx n,m m
0
Now we suppose that the observable is O(x) ≡ ∂x F (x). Integrating by
parts taking into account the PBCs, and then imposing F (x) ≡ x, we get
hχ(xf , tf )|1|φ(xi , ti )i = ~i hχ(xf , tf )|p̂x − xp̂|φ(xi , ti )i, where the operator p̂
is defined analogously to the Hamiltonian operator Ĥ. For arbitrary initial
and final periodic states φ(xi , ti ) and χ(xf , tf ), we finally get nothing else
than the usual commutation relation [x, p̂] = i~.
Indeed we have a theory of relativistic waves where QM emerges as a
consequence of information lost due to fast periodic dynamics. When the
periodicity is too fast, the system can only be described statistically since
at every observation it turns out to be in a random phase of its apparently
aleatoric evolution. This is just like observing a timekeeper under a stroboscopic light [12] or a dice spinning too fast to predict the result. In fact,
we know from the ’t Hooft determinism [4] that there is a deep connection
between a particle moving with high frequency on a circle and the QHO.
It is important to note that, in our case the are no local-hidden-variables
being the time a physical variable that can not be integrate out, and being
the PBCs an element of non locality. Therefore we conclude that the present
theory is not constrained by the Bell’s or similar theorems, and we can talk
about determinism.
3
We are not considering the unimportant phase factors in front of the wave
functions [4]. They shift the energy spectrum giving rise to the vacuum energy [5].
5
3 AdS/CFT interpretation
To extend our theory to interacting fields we could, for instance, develop a
perturbation theory from the PI eq.(2). However, in simple cases, an approximative description of interactions can be given in an easier way. The trivial
case is Compton scattering e′ + γ ′ → e′′ + γ ′′ where we just rewrite the
conservation of the four-momentum among the quanta p̄µγ′ + p̄µe′ = p̄µγ′′ + p̄µe′′
as the conservation of the reciprocal of the four-radiuses 1/Rγµ′ + 1/Reµ′ =
1/Rγµ′′ + 1/Reµ′′ . We see from this example that during interaction the compactification radiuses are subject to deformations. Therefore we face the interaction problem using classical field theory in curved space-time.
For instance, we know from the Bjorken model that the Quark-Gluon
Plasma (QGP) can be approximated as a volume of massless fields at high
energy, and that the energy decays exponentially radiating hadronically and
electromagnetically [13,14]. Using our terminology we get that the periodicities of the fields have an exponential dilatation described by dt2 → dt̃2 ≃
e2ks̃ dt2 and dx2 → dx̃2 ≃ e2ks̃ dx2 , so the QGP freeze-out is encoded in a AdS
metric dS 2 ≃ e−2ks̃ (dt̃2 − dx̃2 ) − ds̃2 . Assuming dS 2 ≡ 0 the proper time acts
as a virtual XD. The propagation of 5D YM fields with a 5D coupling g5 in
such a warped background gives a classical correlator Π(q 2 ) which, in first approximation4 and assuming 1/kg52 = Nc /12π 2 [15,16], can be matched to the
−q2
2
2
quantum vector-vector two point function of QCD: Π(q 2 ) ≃ 2kg
2 log q /Λ .
5
Hence, we obtain classically the quantum runnings of the strong coupling in
agreement with the interpretation [17] of the AdS/CFT conjecture [18].
4 Conclusions
In conclusion, by imposing to a bosonic field the local and dynamical spacetime periodicities 2πRµ ≡ h/p̄µ conjectured by de Broglie, we find that the
theory is in agreement with SR. Remarkably, from the consequent property
of recursivity which is induced to the relativistic time -whose flow as we know
depends on the energy-, the usual QM emerges under many of its different
formulations and for several nontrivial phenomena [1]. This could open a
new scenario where SR and QM can be unified in a deterministic wave theory. After all, the notion of time is strictly related with the assumption of
periodicity: time is defined by counting the number of cycles of phenomena
supposed to be periodic. Therefore, “we must assume, by the principle of sufficient reason”5 , periodicity to define a relativistic clock. In conclusion, since
the whole information is contained in a single period, every single elementary
field can be regarded as in intrinsically compactified space-time dimensions.
4
To evaluate the low energies effective propagator we use the holographic method
with Neumann BCs at the UV scales Λ and boundary field Aµ (q) at the IR scale
µ, in the hypothesis of Euclidean momentum Λ ≫ |q| ≫ µ.
5
“By a clock we understand anything characterized by a phenomenon passing
periodically through identical phases so that we must assume, by the principle of
sufficient reason, that all that happens in a given period is identical with all that
happens in an arbitrary period.” A. Einstein [19].
6
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