Generalised Heisenber Principle

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Comments on a generalisation of Heisenbergs

uncertainty principle and eventual consequences to

phenomena in low energy physics

June 15, 2012

Abstract

In these notes we discuss a generalization of the Heisenbergs uncertainty principle

motivated by some attempts to reconcile quantum mechanics and general relativity.

1 Introduction

Following the traditional way for new developments in physics, quantum theory has appeared

as an explanation for the data from experiments probing Nature at a distance of the order of1010 meters. Its huge success made the quantum aspects of the laws of physics a fundamental

part to understand the World: the phenomena we observe and things that we measure are

though to have their roots in the micro-scale, and even theories describing Nature in bigger

length scales (>1010m) are expected to have their quantum versions.

Quantum mechanics (QM) was extrapolated to a quantum theory of fields (QFT) which,

with the standard model (SM) produce spectacularly good predictions, being the most reliable

theories we have when gravitational effects are negligible.

On the other hand, our understanding of gravity is expressed by Einsteins general rela-tivity (GR) which establishes the relation (equality) between the gravitational field and the

spacetime geometry.

The picture of the World we have is sustained by these three pillars: QFT, SM and GR.

All the rest is speculation. However, this picture is both (a) fragmented and (b) incomplete,

from the theoretical point of view. Since Newtons observation that the force that keeps our

Moon in orbit is the same that makes an apple falls down, physicists are always looking (and

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finding) ways of unifying theories. The lack of an unified theory for all fundamental forces

is therefore disturbing and is what makes this set fragmented. The incompleteness comes

from the fact that against the path followed by our description of the other three fundamental

forces (electromagnetic, weak and strong), gravity refuses to fit into any quantization scheme

known so far.In the regimes of validity of QM and GR there is no evidence of any observed phenomenon

that clearly contradicts them. The knowledge weve accumulated about these theories corrob-

orated by experimental observations is large but an explanation for what happens when we

put both together is still missing. Already in 1916 Einstein pointed out that quantum effects

could modified his theory of gravitation, and in 1927 Klein suggested that a quantum theory

of gravity would modify the concepts of space and time (c.f. [1] and references therein ).

Quantum mechanics has two main features namely the transition amplitude, and the dis-

cretecharacter of its physical content. Since GR implies that gravity = geometry, the quan-

tization of gravity would be related to the discreteness of lengths, areas, volumes, angles,

etc.

In our day-by-day quantum mechanics, generally the only observable with a geometrical

meaning we have to deal with is the position operator. Roughly speaking it is the non-

commutativity of the (complementary) operators the responsible for the discrete spectrum of

the observables in the quantum theory. To this non-commutativity it is associated an uncer-

tainty principle (see equation (A.19)), and in the case of the position operator (whose com-

plementary variable is the momentum) we have the Heisenberg uncertainty principle (HUP)

xipj

2ij ,i, j = 1, 2, 3. Looking at this relation we do not see any reason for the lengthto assume discrete values: one can take the uncertainty in the position (continuously) as small

as needed, paying the price of loosing precision in the knowledge of the momentum.

Let us now consider a simple calculation using the HUP. The reader must keep in mind that

the following argumentation is in some sense quite naive, and cannot be taken in a rigorous

way. As showed in (A.1) the time t required for the expectation value of an observable to

change by an amount equals to its uncertainty is constrained with the change in the energy of

the system by the relation Et 2 . Take the system to be a relativistic particle for which

the uncertainty in the energy1 reads Mc2. Now, suppose all this mass (energy) decays

by light emission2, and therefore the uncertainty in the life-time of this system t can be

replaced by the uncertainty in the position of a photon x divided by the velocity of light.

The uncertainty in the position of the photon must be of course bigger than the Schwarzchild

1For a relativistic particle E= Mc2 and the uncertainty is in fact associated to M.2Conservation of momentum implies the emission of two photons. We shall be sloppy and ignore this,

which might introduce factors of 2.

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radius associated to the mass M, RS = 2GM

c2 , otherwise we would not be able to observe

the photon, since it cannot scape the black-hole. This leads us to conclude that x Lp

whereLp=

Gc3

is called the Planks length.

Notice that the HUP, as said before, does not predict any minimum length, i.e., there is no

limitation in x. However the above calculation shows that when gravity enters in the game

such a limitation appears naturally: HUP breaks down for higher energies, or equivalently

smaller distances (of the order of Planks length) where quantum gravity effects would be

relevant. Of course, the HUP must be recovered when gravity is negligible, so such quantum

gravity effects should be treated as corrections.

In 1989 Amati, Ciafaloni and Veneziano suggested [2] a generalized uncertainty principle

(GUP) in the context of string theory of the form xp 2

+(p)2. The fact that this

GUP implies a minimum length scale can be seen as follows. Without the second (new) term

one can make the localization in position as small as one likes by making the uncertainty inthe momentum large. However, when doing that the RHS grows faster (quadratic in p) than

the LHS (linear in p), and for large enough p, the inequality sign would be contradicted.

To avoid this a minimum and finite xappears, implying a maximum value of p, preserving

the inequality.

Solving this inequality for p the condition3 x

2must be imposed in order to

guarantee real roots. This is, in fact, what defines the minimum value of x. Then by

comparison with our previous discussion we set =

L2p where is dimensionless.

Notice that the correction to the HUP is proportional to the square of the Planks length,

so, it is related to Planks area instead. On the other hand, it is the square of the Planks

length that is proportional to the gravitational constantG, so this correction does look like a

correction dua to gravitation. Nevertheless one could consider a generalization such that the

correction is in Planks length instead. In fact this issue was discussed in[3] and the lesson

we learn from there is roughly that since these terms constitute corrections to a relation that

one still cannot verify empirically, any leading order is acceptable.

Modifications of HUP appear in many places, one being slightly different from another, and

the reasons for these changes are also diverse: from string theory, cosmic rays related issues[4],

modifications of Lorentz symmetry [5], and so on. As we mentioned in the beginning, QM andGR are very well tested theories and therefore their predictions are very reliable. So, among

all the possible places a generalization of the uncertainty principle can come from, probably

a model independent circumstance based on these two theories would be very interesting.

Indeed, Maggiore showed [6]that a different version of the Heisenbergs microscope gedanken

3 0.

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experiment where gravitational effects were included gives rise to a GUP just like that one we

discussed above (see also [7]): xp 2 + (p)2.

The laws of the quantum World produce results that are very distant from what we can

intuitively guess, being completely disconnected from our sensitive experience as humans. In

particular the implications from the uncertainty principle (whose mathematical structure issurprisingly very simple) require reflection, since we would like to understand from where one

should expect some testable modification of it.

The stability of the atom was a huge problem before quantum theory: electromagnetism

predicts that the electron moving around the proton (forming the hydrogen atom) should

radiate, and therefore its motion would be a spiral going towards the nucleus, making our

existence a mystery. The fact that there is the HUP in quantum systems is what makes the

atom stable. First of all, its the uncertainty principle together with the fact that the electron

is confined that makes the electron moves! This can be seen as follows: take the H-atom to

be a box of size 1010 meters. Then, if we want to localize the electron within this precision

(x 1010m) we have an uncertainty in its momentum of p 1024Kgms . The mass ofthe electron is approximately 1030 Kg, so that its velocity is more or less one third of the

velocity of light. Basically when we try to confine (localize) quantum particles they start to

move like crazy. From here we see immediately that its just not possible for the electron to

be at r= 0, i.e., to fall into the proton, and be at rest there (p= 0).

Using the HUP its not difficult to estimate the minimum energy the electron must have

in the hydrogen atom (the binding energy). So, here it is a good place to estimate how

the GUP would change that. The calculation is simple. First, by symmetry we expect thatr = 0 = p, which means that (r)2 = r2, and analogously for the momentum. The totalenergy is given by E = p

2

2m e2

r , and the average of that is thereforeE = (p)2

2m e2 1

r.

Now, HUP implies that p r

, so the mean energy becomesE 22m(r)2

e2r

, and the

RHS of this expression is exactly the minimum energy, so that its derivative with respect to

r must vanish, which give us the value of the radius (for which the energy is minimized)

as 2

me2. Finally, plugging this result into the expression for the minimum energy we get

Emin= 12 me4

2 13.6 eV.

We proceed in the same way but now using the GUP4 xp 1 + L2p

(p)2. Solving

this for the momentum uncertainty we get

p x

2L2p

1

1 4L

2p

(x)2

.

4 Take = 1.

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Notice that the root with the plus sign is ignored. This is because we want to recover the

HUP in the limit L2p 0, which is only possible for the root with the minus sign.Next we consider a Taylor expansion of the RHS of the above expression for small L2p (or

G):

p 1x

+ G(x)3

+ 2 2G2

(x)5+ . . .

Up to first order inL2p we have E 12mx e2

x+ G

m(x)4 E(x) then dE(x)

dx = 1

m(x)3+

e2

(x)2 4G

m(x)5 = 0 which give us the polynomial equation me2(x)3 (x)2 4G = 0.

Notice that for = 0 this is the same equation one gets (in the same step of calculations) for

the HUP. So, we expect the solution to be that solution, i.e., x 1me2

plus something with

. On the other hand, looking at the cubic equation above it is clear that the solution must

be a linear combination of the two sets of constants appearing: x= Ame2 + BG, otherwise

there is no hope to arrange things on the LHS to give zero. Of course, by simple comparison

we can guess that A= 1m2e4

. This, andB = 4me2 are found5 when we plug this ansatz into

the equation and look separately to each order in (i.e.,0 and1). So, x= 1me2

+4me2G.

Now, put this intoE and expand up to first order in to get

E me4

2 + Gm3e8

from where we recognize the usual binding energy of 13.6 eV and a correction which is foundto be of the order of 1048 eV.

This is very small. In [3]it is showed that a contribution of the order of 1023 eV can beobtained if one considers a GUP whose leading order is in Lp instead ofL

2p. However, this is

still 12 orders above any laboratory capacity.

Indeed, it is very important to search for a physical system that could give any measurable

correction predicted by a GUP. A series of papers by Saurya, Elias and others are exactly

in this direction: using a GUP that, according to them, fits in many of the possibilities

already discussed by other authors, they have calculated the modifications in some observable

quantities in well known quantum systems.

In the rest of these notes we shall review particularly two papers, namely, Plank scaleeffects on some low energy quantum phenomena [8] and Discreteness of Space from the

Generalized Uncertainty Principle[9].

5This equation has two more solutions, but imaginary ones, which are not relevant here.

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2 The generalized canonical relations

So far we saw that apparently to reconcile gravity with quantum mechanics one needs in fact to

change what is the cornerstone of the latter: the uncertainty principle. From theorem (A.19)

it is possible to relate the uncertainty principle between two observables with the canonicalcommutation relation of the associated operators, which is needed for the calculations. The

problem we discuss now is that of finding the commutation relations that give rise to a GUP,

but before that let us take a look in the origin of the standard relation [x,p] =i.

Consider a system6 whose physical state is described by an observer, say O1. A secondobserverO2 is displaced a distance a fromO1. One can imagine that the description7 ofthe physical system given by the two observers will be different in certain ways. In order to

communicate, the first observer must make the boost x x + ato reach the second observer,and the second observer makes x

x

ato reach the first. The crucial point here is that the

space being homogeneous implies that the results of the measurements made on the system

must be the same for the two observers8 | 1| 1|2 = | 2| 2|2.This last statement together with the fact that both vectors |1 and |2 are in the same

Hilbert space lead us to use Wigners theorem to relate them by a unitary (or anti-unitary)

operator. Since the only property we are interested in is the position, let us label the vector

in Hilbert space described byO1 as|x, then, inO2 we have|x + a =T(a) |x. Now, asimple calculation (see appendix (B)) shows that the translation generatorT(a) is unitaryand therefore can be written as

T(a) = eiak. Consider now the position operator x, whose

transformation from one reference frame to the other is given by x xa =T(a)1xT(a). Foran infinitesimal translation: xa = x + ia

k,x

+ O(a2). Since xa= x + awe get

k,x

= i.

As showed in appendix (B), consistency with the results from wave mechanics implies that9

k = p

, so that the standard canonical relation between position and momentum operators is

obtained.

What we conclude from this discussion is that HUP is intimately related to the homogeneity

property of the physical space. Specifically, it is this property that implies that measurements

(of the same observable) done by different observers would agree, and as a consequence, the

momentum operator is the generator of the translation transformation.

Apparently it is not trivial to do such a construction based on symmetries to go beyond

6For simplicity we consider only one dimension.7The two observers use the same rules to associate the physical state with vectors in the Hilbert space.

The Hilbert space will be the same for both, but the vectors will not.8The notation is obvious, but to be sure no doubts remain: the index stands for which observer is describing

that state.9Where p= ix.

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the HUP in the direction we pointed out in the introduction. What we can do concretely is

to find a commutation relation between position and momentum operators that can give the

desired GUP.

As a warm up exercise consider the commutation relation [x,p] =i(1 + p2) in the RHS

of (A.19). What one gets is the uncertainty principle xp 2(1 + (p)2 + p2) which(up to this last term that is a constant) is of the kind of GUP we were discussing so far. In [8]

the canonical relation proposed between position and momentum contains probably the most

general polynomial expression in the momentum up to second order:

[xi, pj] =i

ij+ ij1p + 2

pipjp

+ 1ijp2 + 2pipj

; i, j = 1, 2, 3 (2.1)

wherep2 =

3i=1pipi. The constants 1,2,1 and2 can be fixed using the Jacobi identity

[[xi, xj] , pk] + [[xj , pk] , xi] + [[pk, xi] , xj] = 0

as we now show. First, notice that space is still commutative (sorry Paulo!) and therefore the

first term above vanishes ([xi, xj ] = 0) leaving us with

[[xj , pk] , xi] [[xi, pk] , xj] = 0.

Then, using (2.1) we calculate the first term, and for the second we just exchange i and j.

Lets do it bit by bit:

[[xj , pk] , xi] =i

jk1[p, xi] 1

+2pjpkp

1, xi

2

+1jkp2, xi

3

+2[pjpk, xi] 4

.Each of these commutators is calculated under approximations to first order in momentum.

Also, the first two of them require some clever tricks. For the first one we use the following:

from one hand we have10 [p2, xi] = 2[pj, xi]pj = 2i(pi+ 1pip + 2pip)+higher order terms,simply by using11 p2 = pipi. On the other hand, one can write p

2 = p p, so that [p2, xi] =2 [p, xi]p. Then, we compare these two results, which must be the same thing, to get [p, xi]p=

ipi(1 + (1+ 2)p). Multiplying byp1 results in

[p, xi] = ipip1 i(1+ 2)pi. (2.2)10Notice that [xi, pj ] is a function of momentum according to (2.1), so it will commute with any other

function of momentum.11In most places we use Einsteins sum convention.

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Commutator labelled as 2 ispjpkp

1, xi

= pjpkp1, xi

+pj[pk, xi]p

1 + [pj, xi]pkp1.

The second and third terms on the RHS are given by (2.1). The first one must be found,

again, using a trick, which consists in writing 1l = p p1, then [1l, xi] = 0 becomesp [p1, xi] +[p, xi]p

1 = 0, from where we find using (2.2)p1, xi

= ipip

3 + i(1+ 2)pip2. (2.3)

At this point the commutator number 2 ispjpkp

1, xi

= ipjpkpip3 + (1+ 2)p

2

iik+ ik1p + 2pipkp + 1ikp2 + 2pipkpjp1 i

ij+ ij1p + 2

pipjp

+ 1ijp2 + 2pipj

pkp

1. (2.4)

Now the commutator number 3 is direct: [p2, xi] =2pj[xi, pj ]. Using (2.1) and organizingits terms we get

p2, xi

= 2ipi+ (2+ 2)pip + (1+ 2)pip2 . (2.5)Finally the last of the four commutators is also direct:

[pjpk, xi] = ipj ik+ ik1p + 2pipkp

+ 1ikp2 + 2pipk

ipk

ij+ ij1p + 2

pipjp

+ 1ijp2 + 2pipj

. (2.6)

The next step is to plug results (2.2), (2.4), (2.5) and (2.6) back where they belong and rewrite

the Jacobi identity:

1p

1 + 1(1+ 2) 2p1 21 21p + 21+ 21(1+ 2)p+ 21(1+ 2)p

2 2 21p 12p2 (jkpi ikpj) = 0.Ignoring the higher order terms we find

(1 2)p1 + 21+ 21 2= 0.

Now, taking1 = 2 = aand (for dimensional reasons) 1 = a2 this equation fixes2= 3a2

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and we are left with equation (1) of[8]:

[xi, pj] =i

ij a

ijp +

pipjp

+ a2

ijp

2 + 3pipj

, (2.7)

witha= a0Lp where a0 is dimensionless.

As discussed previously in ordinary quantum mechanics the momentum operator is found

to be the generator of translations in space, and in the space of wave functions it acts as a

derivative, while the position operator acts multiplicatively. In fact this can be easily verified

from the HUP: [x, ix] = i. Such a simple representation is quite unlikely to be foundfor the momentum operator satisfying the GUP (2.7). On the other hand we have to keep

in mind that when gravitational effects become negligible we need to recover HUP from this

GUP, and in the same way we need to recover that usual representation of the momentum

operator as a derivative. So, thinking in the context of perturbations what we12 do is to

consider the momentum operator to be

pj = p0j+ Ap0p0j+ Bp20p0j

where p0j =ij, is the usual momentum operator and p20 =3

i=1p0ip0i. The position

operator is the same as the usual (notice that the RHS of the GUP contains no position), so

[xi,p0j] =iij. Naturally we expect the coefficientsA and B to be related to a, so that when

G 0 we get pi p0i. In fact, from dimensional analysis we need A aand B a2.

In order to fix the coefficients we plug the above formula into [xi, pj], and then compareto (2.7). So,

[xi, pj] =iij+ A [xi, p0p0j] + B

xi, p20p0j

.

For the first term on the RHS we have

[xi, p0p0j] =iijp0+ [xi, p0]p0j

and we need this last commutator. In order to find it we use a similar trick as used to find

(2.2). First of all we have [xi, p20] = 2[xi, p0k]p0k = 2iikp0k = 2ip0i(p0p

10 ) 1l

. But the LHS

can be written as 2 [xi, p0]p0, which implies that [xi, p0] =ip0ip10 . Then

[xi, p0p0j] =iijp0+ ip0ip10 p0j.

Now, this last term can be improved. Up to first order in awe havepj p0j(1 + Ap0)+O(a2).12The reader has to remember that most of the we in these notes refer to the work of Saurya and others.

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Thenp2 =pjpj =p20 + 2Ap

20 + O(a2), so that to order zero in a we getp p0 and also, to first

order ina pj p0j(1 + ap). This last relation can be inverted and once again approximated upto first order inagivingp0j pj(1 ap). With this result we can calculatep0, which appearsin the first term on the RHS of the commutator we are working in: p0 =

p0jp0j p(1 ap).

Also, the product we are improving becomes p0ip1

0 p0j pipjp1

(1 ap). Finally[xi, p0p0j ] =iijp(1 ap) + ipipjp1(1 ap). (2.8)

The second commutator isxi, p

20p0j

= 2p(1 ap)ipipjp1(1 ap) +p2(1 2ap + a2p2)iij. (2.9)

Considering the canonical relation up to second order in a and in the momentum we have

[xi, pj] iij+ Apij+ Apipjp1 + 2B A2pipj+ (B A2)p2ijand when compared to (2.7) gives A = aand B = 2a2, leading us to equation (5) of [8]:

pi=

1 ap0+ 2a2p20

p0i. (2.10)

3 The generalized Schrodinger equation

The quantum state evolves accordingly to the Schrodinger equationH = it, where thehamiltonian operator isH= p2

2m+V. Following this perturbative philosophy we should expect

our hamiltonian to be something likeH =H0+aH1+a2H2. Indeed, this is obtained whenwe square the momentum given by (2.10): p2 =p20 2a|p0|p20+ 5a2p40.

Here we point out that the operator with third power in momentum is not well defined13.

Once the Schrodinger equation is a scalar equation, all momentum operators appearing there

are in fact related to the norm|p0|. Naturally, the occurrence of a square-root in odd powersof this quantity make them untreatable by the usual means.

Ignoring the fact that we dont know how to act with this third power in momentum

operator, and for convenience writing it as3, the generalized Schrodinger equation (GSE)reads

it=

2

2m2 + V ia

3

m3 +5a

24

2m4

. (3.11)

We might think of this equation as a correction to the Schrodinger equation when gravitational

13Only in one dimension we can in fact define it since there a scalar and a vector are the same thing.

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effects are taken into account.

3.1 Discretization of space

In 2009 Ahmed, Saurya and Elias [9]considerer non-perturbative solutions of the one-dimensionalgeneralized Schrodinger equation (3.11) up to first order in a for a free particle of mass M

confined to a box of sizeL. As usual they showed that for the stationary case energy is quan-

tized, but also a new phenomenon takes place: the size of the box the particle lives cannot

be anyone; it must be a multiple of the Planks length. In what follows we reproduce theses

results.

Consider the GSE

it= 2

2M2x+ V

ia3

M 3x (3.12)

whereV is that of a square-well

V =

0 if 0 x L; otherwise.

Notice14 that although here one can define the cubic momentum operator, the parity trans-

formation is no longer a symmetry of the equation. This is quite unpleasant since space is not

isotropic in that case.

For definite energy configurations we plug the ansatz (t, x) = (x)eiEt in (3.12) ob-

taining

d2

dx2 i2ad

3

dx3 =

2ME

2 .

Define k2 2ME2

and consider (x) emx. The above differential equation becomes apolynomial equation inm:

2iam3 + m2 + k2 = 0.

Fora = 0 this problem is the well known quantum-well problem, and the solution is m = ik.Now, for a= 0 we expect some generalization of that. Calculating the solutions of thisequation and expanding up to first order in a we get

m= ik(1 + ak) m= ik(1 ak) m= i2a

i2ak2,

(x) =Aeikx + Beik

x + Ce( i2a

i2k2a)x,

14This remark appeared in one of the many fruitful discussions with Rodrigo Pereira.

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As we said, for a 0 we must recover the result of ordinary Schrodinger equation. Since thelast term of the expression above cannot be expanded, we require C 0 whena 0.

The boundary condition(0) = 0 implies a constraint between the constantsA+B+C= 0,

so that we take B = (A + C). Then, the above solution is written as

(x) = 2iA sin(kx)eik2ax Ceikxeik2ax + Ce i2axei2k2ax.

Lets take A= i B2, so that for a = 0 we recover the usual solution of a quantum particle ina box(x) = B sin(n

Lx). Then we expand what we can in the above equation, obtaining

(x) B sin(kx) + Ce i2ax Ceikx + ik2ax

B sin(kx) Ceikx 2Ce i2ax

.

Now, since|C| 0 for a approaching zero, we expect that|C|depends at least linearly in a.

This means that the two last terms on the RHS of the above expression can be neglected forthey are at least of second order in a.

Then, we consider the other boundary condition(L) = 0 usingC= |C|eiC. This givesthe relation

B sin(kL) = |C|

ei(kL+C) ei( L2aC)

ik2aLB sin kL.For a0 we have k n

L (n ZZ) and|C| 0. Then, the LHS above vanishes as well as

the RHS. For the case a = 0 we take kL = n+ with IR going to zero for a 0. Thenthe factor k2a falls to zero faster than a, and we can ignore that last term in the expression

above. We remain with

B sin(kL) = |C|

ei(kL+C) ei( L2aC)

,

whose imaginary part is

sin(kL + C) + sin

L

2a C

= 0

that is satisfied for L2a

C= 2q (n+ C) and L2a = (2q+ 1)+ n+ 2C, with q ZZ.

This shows that the length of the box obeys a discretisation rule15

and therefore cannotbe anyone! This is in agreement with our first discussion in the introduction of these notes:

we expect that quantum gravity implies discretisation of geometry.

15Notice that the parameter Cis not determined but is fixed.

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3.2 Effects of GUP in the quantization of the magnetic flux in a

superconductor

Here, as in[8], we shall not go deep in the theory of superconductivity (it is pointless). The

only feature we will need is the quantization of magnetic flux that happens in superconductingmaterials. Then, we shall evaluate how the generalization of Heisenbergs uncertainty principle

(2.7) would change that.

The superconductivity phenomena is characterized, among other things, by the fact that

the sample behaves as if it had no measurable electrical resistivity. Due to interactions with

the atoms of the lattice, a tiny attraction appears between two electrons moving in the sample,

and they can form a bound state16 which will be described as a Bose particle of charge 2e.

These pairs, called Coopers pairs, exist for very low temperatures, and any increase of it can

destroy them. The wave function of the Cooper pair is defined by the Schrodinger equation

of charged particles moving in a magnetic field17 (see appendix (C)):

it= 1

2m(i qA) (i qA) . (3.13)

The localU(1) symmetry of this theory implies, due to Noethers theorem, a conserved charge

given by (the details are presented in appendix (C))

dx ||2, whose associated current is

J= 1

2m(D+ (D))

with D= i q.Now if we write the wave function as =

ei a direct calculation gives

J=

m

q

A

.

Let us consider a sample which is a lump. This (the topology of the sample) is very important.

Notice that the densitymust be uniform. If this was not the case a net charge would produce

an electric field pushing the electrons apart and destroying the Cooper pairs.

Then, taking the divergence of the above equation, and using the gauge where A= 0,one finds that in the steady state 2= 0, which is satisfied for a constant phase . This alsogives J A, with proportionality constant q

m.

16Thanks to the remaining N 2 electrons as showed by Cooper.17We wrote q= 2e instead.

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Locally the magnetic field can be written as B= A. From the equations

B 00tE= 0J E + tB= 0

with J=

q

m

A we get London equation

2A=

q

0mc2

A.

Just to get a hint about what is happening here one might consider this equation in one

dimension. The solution is of the form ex , with18 2 = q

0mc2 and we see that the field

decreases when we go inside the material. Thus, the magnetic field does not penetrate the

sample very deeply. This is called the Meissner effect: if one takes a magnetic material at high

temperature, then apply an external magnetic field to it, and finally lowers the temperature

below the critical temperature19

,the field is expelled.Now, suppose that the sample is, instead of a lump, a ring with thickness larger than .

What we are going to do is the following. First, at high temperatures, a magnetic field is

applied. Then, we cool down the sample below the critical temperature, which makes the

material a superconductor, and as we saw, the magnetic field is expelled from it. The final

step is to remove the source of magnetic field. However, from E+tB = 0, and fromthe fact that there is no electric field inside the superconductor we conclude that the flux

of magnetic field cannot change after we remove its source: ddt

= 0. In order to solve this

apparent contradiction Nature bends the lines of field trapping them to the ring. After that,

a current in the surface of the ring is generated to keep the flux constant.

Remember that for a sample which is a lump, we took to be a constant. However, this is

not the case for our new sample, which is a ring. Since inside the superconductor J= 0 one

gets =qA. The flux of magnetic field inside the superconductor is (, B) = A dl,then = q. Due to the external gauge field one cannot expect to get = 0. Imposing,

however, that the wave function is single valued (and not ), i.e., that = 2n we find out

that

nh= q n ZZ

concluding that the trapped flux is quantized, as multiples ofh/q.

It is clear now that since the GUP implies a modification of the Schrodinger equation the

electrical current predicted from it will be different and ultimately we might have a contribu-

tion to the quantum of flux. This is what we need to calculate now.

18This characteristic length is called London penetration depth.19The critical temperature is that for which below it some materials become superconductors. Measurements

go from miliKelvins to 20 Kelvins.

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As we mentioned before the correction to first order in a to the Schrodinger equation gives

a ill-defined operator (at least using the proposed representation for the momentum ( 2.10)).

Then, we shall ignore such a term and consider the leading order to be in a2.

However when doing that we cannot simply ignore the problematic term and use what is

left in (3.11). It is necessary to reconstruct a generalised Schrodinger equation from a GUPwhose leading order for the corrections is a2. This can easily be done following the same

scheme used before and the result is

[xi, pj] =i

ij+ ija2p2 + 2a2pipj

. (3.14)

Then, the momentum is given by

pj =p0j+ a2|p0|2p0j

and the generalized Schrodinger equation

it= 2

2m2+ V + a2

4

m4. (3.15)

From this equation we get the expression for the electrical current, which now has two parts

J0 and J1, like in (D.26) and (D.27), but with the minimal coupling. In particular the

contribution to the quantum flux comes from

Jgauged1 =

1

m i qA i qA2

+ i qAi qA

2

+

i qA

i qA

2+

i qA

i qA

2

.

In order to estimate that we notice that after the minimal coupling is considered in ( D.27) we

get lots of iqAterms. Since for the superconductor the density = ||2 is uniform the

first term involving the gradient of the wave function is simply . The flux of magneticfield is given by

A dl that can be approximated to|A|2R, with R the radius of the

sample (a ring). So, we see that the flux depends on the typical size of the sample. Then, theorder of magnitude of this first term involving the change of the phase can be found taking

hR

. For the second term we consider20 |A| 12|B|R, and take q =2e, remaining

with q|A| e|B|R. Considering the following values for the size of the sample and thestrength of the magnetic field as R 103m and|B| 103T, we see that the first term

20Since B= A we get A= 12B r.

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is 6 orders of magnitude smaller than the second one. Then, equation (3.16) is simplified to

Jgauged1 32e3|A|2A. Then, the total current is

Jgauged =

m+2e

mA + a2

32e3

m |A|2A

.

Since the circulation of it vanishes inside the sample we get

2e=

1 + 16e2a2|A|2where =

A dl. Imposing that the wave function is single valued fixes = 2n, with

n ZZ. Then we can invert this relation by doing an approximation to order a2 obtaining

=n0

1 16e2a2|A|2

where 0= h2e .

Now in order to estimate the parameter a0 in the definition ofa we notice that the de-

viation from the usual quantum flux given by 0 = 16e2a2|B|2L20 cannot be detected or

distinguished from what we really measure. So, supposing that indeed what we measure is a

flux quantum plus a correction due to effects in the Plank scale we can blame our technologi-

cal limitation and say that the quantity 16e2a2|B|2L2 is less than our experimental error 00 ,which we can take to be 1 part in 10. Using the values for the sample size and magnetic field

used before we get

a0 10262 ,

and, of course, with better precision (much better than that we have nowadays) we could have

hope in detecting the two contributions to the flux of magnetic field.

3.3 The Lande g-factor in the non-relativistic approximation using

a GUP

Consider Diracs equation for the 4-dimensional spinor

it=

c p + mc2where

i =

0 i

i 0

i=

1l 0

0 1l

are 4 4 matrices, i,i= 1, 2, 3 being the Pauli matrices.

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Using equation (2.10), up to first order in a we get p = p0 ap0 p0, and fromp0 (1 ap)p, we approximate p0 p, and replace it by p getting the generalised Diracequation (GDE)

it=

c p c a( p0)2 + mc2

. (3.16)

Turning the gauge field on, and calling the canonical momentum 0 we consider the non-

relativistic approximation writing

(t, x) =

(t, x)

(t, x)

e

imc2t

so that (3.16) can be written as the set of two equations

mc2 + it = c 0 + mc2 + e ca ( 0)2 (3.17)

mc2 + it = c 0 mc2 + e ca ( 0)2 . (3.18)

In (3.18) we suppose a low-varying field t0 and a weak electric field so that21 0.Then it becomes

c 0

2mc2 + ca( 0)2

.

This can be approximated to

12mc

0

1 a( 0)2

2mc

Plugging this result back into equation (3.17) we get

it

( 0)22m

a

c( 0)2 +( 0)4

4m2c

,

i.e., a corrected Schrodinger equation due to Plank scale effects.

Working on the kinetic terms using ij = 1lij+ iijkk, and defining the operator S= 2

we get

it = 12m

ca2 a4m2c

4 2 e2mc

1 2acm amc

2S B ae

22

4m2c3|B|2 aie

2m2c2

( B)

e

cA

+ i2( B)

.

We recognize from the third term the gfactor. Using the same reasoning used for the21In fact, the electric field is irrelevant for our discussion.

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superconductivity:

g= g0(1 ag1)withg1= 2mc +

2

mc and g0 = 2. Then ignoring

2

mc compared to mcwe get

a0 < mcLp

10 1020

where 10 is the experimental precision.

A The uncertainty principle and the algebra of the ob-

servables

For completeness we shall introduce some definitions from the statistical (Borns) interpreta-tion of quantum mechanics. In order to do so we simplify the discussion to one dimensional

space and consider the probability to find a point-like particle whose wave function is (t, x)

in the region L.

Suppose a very large number of particlesNall in this physical states(t, x). By measuring

the position of all these particles simultaneously we get nof them inside the region L, and

this is predicted by the probabilistic interpretation as

n

N

=PL+ statistical errors

(t, x)(t, x)L.

Another way to get this information is the following. One could perform an experiment in

which a single particle is placed in the state (t, x), and measure its position. Then, the

identical experiment is repeated N times, so that n would be the number of experiments in

which the particle was observed in the region L.

Now, consider a series of experiments to measure the particle whose initial state is(t, x).

Letxi be the position of the particle measured in thei-th experiment andNthe total number

of experiments. The average value of the particles position will be then x= 1N

Ni=1 xi. The

average square deviation is defined as (x)2 = 1

NNi=1(xi xi)2 = x2 x2 and the root meansquare, x= (x)2.The numbers xand x can be reported as results from experiments,and therefore constitute important informations about the system.

Suppose now we make a series of measurements on the quantity A, which can only take

the values a1, a2, . . . , an. The corresponding probability to find these results are p1, p2, . . . ,

pn. Then, the expectation value ofA isA =n

i=1 aipi.

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On the other hand, the average value ofA is A= 1ni=1Ni

ni=1 Niai but sincepi=

Nini=1Ni

we getA = A.With that we can rewrite our statistical quantities using the Borns interpretation. The

probability of finding the particle inside a region dx is Pdx(x) = (t, x)(t, x)dx, so we get

x = + xPdx(x), andx2 = + x2 Pdx(x), and therefore(x)2 = (x x)2 = x2 x2

so the uncertainty in the measurement of the position of the particle is defined by (the root

mean square)

x=

x2 x2

When we measure the observable A, its value is determined and therefore A = 0 at

least at the instant of the measurement. Define the hermitian operator

22aA A, andthen (A)2 =|a |a =a| a, where|a =a |. Now since A = 0 and (A)2 =a| a 0 for all x we conclude that|a =a | = 0, soA | =A |, i.e., any statewith vanishing uncertainty in the observable Ais an eigenstate of the corresponding hermitian

operatorA.If we want to measureAandB simultaneously, then the physical state has to be eigenstate

of bothA andB, which is possible only if A,B = 0. Otherwise we have the followingtheorem:

(A)2

(B)2

A,B2i 2

(A.19)

whose proof is now presented.

To the observable A it is associated the hermitian operatorA with uncertainty A =A2 A2 in the state|. Define the hermitian operatora=A A so that (A)2 =

| (A A)2 | =a2. Define the state|a =a. Now, letB be another hermitianoperator and use similar definitions as those forA. Then

(A)2(B)2 =

|aa | |bb | = a| a b| b = |a|2

|b

|2.

Using the Schwartz inequality we get

(A)2(B)2 | a| b|2 = |ab|2.

This last term is the norm of a complex number, which can be separated into its real and

22a =a.19


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imaginary part, giving: |ab|2 =

Reab2

+

Imab2

. Since the real part is always

bigger or equal to zero the following inequality holds: |ab|2

Imab2

. Finally, writing

this imaginary part as Imab = abab2i

= abba2i

= [A,B]

2i we conclude the proof of the

theorem.The HUP is found from the quantization ansatz [x,p] =idirectly.

A.1 The time-energy uncertainty

As a consequence of the theorem (A.19) we show that Et 2 .

Consider the observable A, and suppose that after a time t,A changes by A. Fromthe formulas showed in the previous section for the average of an observable a direct calculation

making use of the Scrhodinger equation gives

d

dtA = d

dt

dx A= 1

i

A,H .Then, from our hypothesis ddtA

t= Aand therefore

1

A,

H

t= A.

Now, using (A.19) we get 1

AE 12

A,H= 12At

so, Et 2

.

B The translation operator

Consider a state|x and the transformation|x + a =T(a) |x. Now, lets show that thetranslation operatorT(a) is unitary. Take

x

|T(a) |x = x| x + a =((x + a) x) = x a| xwhere in the last step we used the fact that the delta function is symmetric. Then, x|T(a) =x a| and we conclude thatT(a) =T(a), so the adjoint of the translation operatortranslates backwards. Moreover it is direct to see thatT(a)T(a) |x = |x, i.e. this operatoris unitary and therefore can be written asT(a) =eiak wherek =k.

Consider the eigenvalue problem k |k = k |k, where|k are also the eigenstates of

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T(a). Take the projection of the translation operator onto the position space:x|T(a) |k =eiakk(x). Doing the same thing but now usingT(a) =T(a) we getx|T(a) |k =x a| k = k(x a), and comparing both results: (x a) = eiak(x). Expanding theRHS for small a up to first order and after that taking the limit a 0 we getix =k.

Using the de Broglie relation p= k and taking p=ix we arrive to the conclusion thatk= p

.

C The minimal coupling in electromagnetic phenomena

Roughly speaking the gauge principle states that when we promote a global symmetry of the

theory to become a local one, it might be necessary the introduction of new fields - the gauge

fields.

We start by writing the wave function in an internal space basis as = aua. The gaugegroup is the local U(1), whose elements are g(x) = e

iq(x), and it acts on the (internal

structure of the) wave function as g. Now, when we go from one point of the spaceto another as x x+dx the wave function changes according to +d, where d =(x + dx) (x). On the other hand,d = daua+ uadua, where this change in the internalbasis elements is given bydua= g(dx)u u. Let us make a quick (and trivial) considerationto get g(dx). First, notice that g(0) = 1l implies (0) = 0. Then g(dx) = e

i(dx), and

(dx) = (0 +dx) (0) +(x)dx, so (dx) = d, and finally g(dx) = eid. So,considering a first order approximation in g(dx) the change in the internal basis becomes1 i

qdua = ua+ dua, which gives dua = idua. Together with that we have da =

adx and at the end: d=

a iq a

uadx

.

Introducing the definitionA(x) (x) (and calling A(x) the gauge field or connec-tion), so that the gauge group element reads g = e

iq

Adx

, and also defining what is called

the covariant derivative as D =

a iq A(x)a

ua, we get to the conclusion that the

wave function changes from one point to the other according to

(x + dx) (x) =D dx.

Another (more practical) conclusion is that the covariant derivative (and not the ordinary

derivative) transforms in the same way as does under the gauge transformation if the gauge

field transforms as A A iqgg1 =A , for g = eiq(x), i.e., D gD when

g.Next we analyse the so called geodesic configurations, defined by D = 0. The wave

functions defined by this equation are such that it is the presence of the gauge field alone that

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makes changes on them. For convenience define e q

, and this equation reads

ieA= 0. (C.20)

The solution of this equation can be found iteratively and is given by

(x, xR) =eieAdx

R

where stands for a path in spacetime joining the points xR and x, and R is the wave

function calculated at xR. For completeness we discuss in (??) how the solution is found.

This derivation is not needed for the rest of the discussion and can be ignored by the reader.

Notice that if we turn off the gauge field then (x, xR) = R, i.e., the wave function at

the final point is the same of that in the initial (reference) point. That is what one expects

since according to equation (C.20), the only thing changing the wave function is the gauge

field, which is not there in this case. So what the gauge principle is telling us is that the wave

function of a free particle differs from that of a particle in an external electromagnetic field

by a phase.

We put a under the wave function symbol in order to stress out that when the gauge field

in on, the wave function will change by a multiplicative phase, but this phase will depend on

the path of the particle. Suppose we solve equation (C.20) for a path 1 going fromxR toxf,

with initial conditionR. Then, imagine we take the same initial condition, and the same end

points, but now a different path, 2. The wave function in the first case is23 1 =e

ie1

AR

and in the second case 2 =eie2 AR. We have seen that two different solutions of equation

(C.20) can be related by a gauge transformation eie(x). Then, suppose this is thecase and1 and 2 differ by this multiplicative phase: 1 =e

ie(x)2 . We then find out

that = 1

12

A, where 1 12 is the loop whose base point is xR. Now, Stokestheorem implies that the integral of the gauge field over this loop equals to the integral of its

curvature F = dA over the area whose boundary is , i.e., the flux of the field strength

through the area : = 1

12

A=

F (, F).We have seen that when a local U(1) transformation is considered the gauge potential

enters in the game and not only the wave function changes by a (local) phase factor but also

the ordinary derivative must be replaced by a covariant derivative. What we need now is

to take all these new features into the Schrodinger equation in order to describe a charged

particle in an electromagnetic field.

The lagrangian of a non-relativistic particle of electric charge e and massm in an electro-

23Some times we might use the differential form notation: = 1n!1ndx

1 dxn .

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magnetic field described by the gauge potentials Aand is

L=1

2mv2 + ev A e. (C.21)

The first step to get the hamiltonian function, and therefore the Schrodinger equation, is to

get the canonical momentum:

p=L

v =mv+ eA + eA (C.22)

where= mv defines the kinetic momentum.

The hamiltonian is defined by

H= (p v L)

v=v(p)and equation (C.22) can be inverted to give the velocity in terms of the momentum24 leaving

us (after a straightforward calculation) with

H= 1

2m|(p eA)|2 + e.

Then, the minimal coupling discussed before has the following practical implications

p p eA H H e,

i.e., the ordinary (kinetic) momentum (which, in fact, should have been represented here by

in the LHS of the first substitution) is replaced by the momentum which has the ordinary

one plus a contribution from the external (gauge) field; and the energy gets also a contribution

from the scalar potential.

We finally arrive to the conclusion that the local gauge U(1) transformations transform

the free particle wave function by a local phase and the free particle hamiltonian as

H= p2

2m H e=|p eA|

2

2m . (C.23)

The first step to build up the quantum version of this theory25 is to promote the variables of

24In the case the magnetic field is too strong so that the kinetic term can be neglected, the equation definingthe canonical momentum in terms of velocity cannot be inverted and we have a constrained system. In thatcase the canonical quantization procedure must be done using the Dirac-Bergmann algorithm, and we end upwith a non-commutative theory (is that right Paulo?).

25Following the canonical quantization scheme

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the phase space, position and momentum, into operators whose representation is

x x= x p p= i

and (also/therefore) [x,p] = i. The gauge field acts multiplicatively and since it is in an

abelian Lie algebra A, A = 0. Now, the quantum hamiltonian is obtained by putting ahat in the quantities at the RHS of (C.23):

H= 1

2m

22 e

iA 2e

i A+ e2|A|2

+ e. (C.24)

Notice that the second term on the RHS can be made to vanish in the Lorentz gauge. The

Schrodinger equation is finally obtained taking H=it.

It is possible (and nice) to formulate this as a field theory for and . The free theory

is described by the lagrangian

L= i

2 (t t)

2

2m.

One can check that taking the variation of the action S = dtL with respect to , andcomparing the result with the equation obtained from (C.24) with e = 0. For pedagogical

reasons lets rewrite the above lagrangian in an apparently inconvenient way, with the is and

together with the derivatives:

L=1

2((it) + (it))

1

2m((i)(i)) .

Although strange, with this expression it is now easier to consider the minimal coupling

i i eA i i eA

and

it it e it it e.The gauged lagrangian becomes

L = 1

2 (it e) +1

2 (it e)

12m

(i eA) (i eA) (C.25)

and one can verify that variation of the corresponding action with respect to gives exactly

(C.24).

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As discussed before the minimal coupling implies that the ordinary derivatives are replaced

by covariant ones, which is exactly what is happening here. One can introduce

D0 = it e D= i eA

and write the gauged lagrangian as

L=1

2D0+

1

2(D0)

12m

D(D)

which is a much nicer and compact form.

C.1 Solution of the holonomy equation (C.20)

Let be a path in spacetime parametrized by x

(), such that x

(0) = xR is theinitial point - called also reference point. We consider the spacetime manifold to be piece-wise

simply-connected, so that the path is defined inside a simply-connected region. We want to

solve (C.20) for configurations on this path. Then, the first step is to project (take the inner

product) this equation on the velocity vector dx

d in each point of . Calling C= ieA for

convenience we getd

d+ C

dx

d= 0.

Integrating this equation in from = 0 to some other point we get

[, 0] [0, 0] = 0

C() dx

d[

, 0]d.

The second term on the LHS will be denoted asR since it is the wave function calculated at

the reference point. Now, on the RHS we have still a wave function inside the integral. Using

again the same equation above we can write this wave function as

[, 0] =R

0

C()

dx

d[

, 0]d

and plug it back getting

[, 0] =R 0

C()

dx

ddR+

0

C()

dx

dd

0

C()

dx

d[

, 0]d.

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We notice that we still have the wave function on the RHS, so we repeat the procedure once

more

[, 0] = R

0

C()

dx

ddR+

0

C()

dx

dd

0

C()

dx

d[

, 0]dR

0

C()

dx

dd

0

C()

dx

dd

0

C()

dx

d[

, 0]d.

and in fact recurrently. A great simplification here is that the connection is abelian and

therefore we do not need to worry about path ordering. The RHS of this expression can be

written as

[, 0] =n=0

(1)n10

dn d1dxn

dn dx

1

d1C1(1) Cn(n)R.

We notice that there is an ambiguity when calculating the integrals. For instance, in the

second term we have 0

dC()

2=

0

10

d1 d2 (C(1)C(2))

and the integration variables can be exchanged without affecting the result. Then we conclude

that

120d1 d2 (C(1)C(2)) =

1

2

0

dC()

2

,

ans similarly for the other terms we get

1

n!

0

dC()

n.

So, the solution is written as

[, 0] =n=0

(1)nn!

0

dC()

nR,

which is formally the series of the exponential function:

[, 0] =eieAdx

R.

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D The current for the generalized Schrodinger equation

Let us consider a practical way of calculating the conserved current associated to the U(1)

symmetry which involves a trick with the equation for and its complex conjugate.

Take the free (with no external gauge field) GSE

it= 2

2m2+ V + a2

4

m4

and multiply it by . Then, consider its complex conjugate multiplied by. Subtracting

the second result from the first we get (writing it in a very pedagogical way for the minimal

coupling to be considered later)

it

|

|2 =

1

2m (

i

)2

(i

)2 +

a2

m (

i

)4

(i

)4 .

Both terms above can be written as the divergence of a gradient, as we now show. For the

first one it is direct to check that it is2 ( ) . For the second one we firstnotice that it appears in

( ( )) = 2 2 2 + 4 4.Now we notice that the first term in the RHS can be written as 2 (2 2),and also we have ( ) = 22+22. Finally,we end up with

4 4 = 2 2+ 2 2 .Now we can write a continuity equation t + J= 0 where = ||2 and

J= J0+ a2J1,

with

J0 = 1

2m

i + i

(D.26)J1 =

1

m

i

i2

+

i

i2

+

i

i2

+

i

i2

.

(D.27)

If one turns the gauge field on the currents above change according to the minimal coupling,

as discussed before.

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Generalised Heisenber Principle

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