Theory of Measurements 6B

We continue our discussion of the measurement of a path PQ being measured with a stack of measurement circuits. The path PQ represents an information

δ

-function which is either singular or complex in nature. The information contained within the Path PQ, is extracted by the observer using M_ckts. If all the information is extracted, the observer has arrived at the point Q from the starting point P. The observer stays in the state of measurement until the measurement is complete.¹

It is important to note that the measurement circuits M_ckt, corresponds to the measurement capability of Obs_1/1 (Obs_c). The system in the state of the measurement i.e. the electron orbiting around a proton, corresponds to the measurement capability of the Obs_1/137 (Obs_elec).

Equivalently, the vector space V corresponds to Obs_1/1, and the covector space V* corresponds to Obs_1/137, such that V*:V

→

ℝ

. We note that Lorentz Invariance / Möbius Invariance along with Locality and Unitarity, is implicit. All the physical measurements, are made and interpreted in S-frame.

Let us quickly review the relativistic wave equation or Klein-Gordon equation first.

We have a rather simple homemade criterion in j-space, based on the fact whether the measurement circuit M_ckt can be completed or not, while making a measurement.

     If the measurement circuit can be completed then we can write a relativistic wave equation for a spin-less particle, known as the Klein-Gordon equation.

    The idea is fairly straightforward. We set up a field based on oscillations (harmonic). We then write Lagrangian density of the field. Next, we use Euler-Lagrange equation to derive the equation of motion, which is the Klein-Gordon equation. An important point to note, is that mass appearing in this equation is based on interactions between oscillators. We provide a quick summary as follows.

Consider a pair of coupled (interacting) harmonic oscillators as shown. We can write the Kinetic Energy T, Potential Energy V, and then the Lagrangian for the configuration as,

T = \frac{1}{2}m{\dot{{x}_{1}}}^{2}+\frac{1}{2}m{\dot{{x}_{2}}}^{2}

V = \frac{1}{2} k x_{1}^{2} + \frac{1}{2} k x_{2}^{2} + \frac{1}{2} κ (x_{1} - x_{2})^{2},  and

L=T-V

=\frac{1}{2}m{\dot{{x}_{1}}}^{2}+\frac{1}{2}m{\dot{{x}_{2}}}^{2}-\frac{1}{2}k{x}_{1}^{2}-\frac{1}{2}k{x}_{2}^{2}-\frac{1}{2}κ({x}_{1}-{x}_{2})^{2}.

Now, suppose we place these oscillators infinitesimally close together i.e.

\left({x}_{2}-{x}_{1}={0}_{j}\right)

. ( The value of the 0_j is based on the measurement capability of the observer Obs_1/1, i.e. The locality condition. Therefore the precise measurement of 0_j is not possible for Obs_1/137, which has a lower measurement capability than Obs_1/1.)

We select coordinate axes as,
(i) x - the separation between the oscillators, and
(ii) ϕ(x) - their respective vertical displacements also known as the field amplitudes.


    Imagine that we have infinite many of these oscillators infinitesimally close to each other. In this case, x is a continuous variable.

And, rather than calculating total energy of the configuration, we write the Kinetic Energy Density

𝒯

, Potential Energy Density

𝒱

, and the Lagrangian density

ℒ

, for the unit area of the configuration.

𝒯=\frac{1}{2}m{\dot{φ}}^{2}

, and

𝒱=\frac{1}{2}k{φ}^{2}+\frac{1}{2}κ{{∂}_{x}φ}^{2}

Substitute

κ=m

and

Φ=\sqrt{m}φ

and then

{𝓂}^{2}=\frac{k}{m}

to obtain,

𝒯=\frac{1}{2}({∂}_{t}Φ{)}^{2}

, and

𝒱=\frac{1}{2}{𝓂}^{2}{Φ}^{2}+\frac{1}{2}{({∂}_{x}Φ)}^{2}

We note that by using the condition

κ=m

, we have made the mass measured in the unit area, purely interaction dependent. We can write the Lagrangian density as,

ℒ(t,x)=𝒯(t,x)-𝒱(t,x)==\frac{1}{2}({∂}_{t}Φ{)}^{2}-\frac{1}{2}{𝓂}^{2}{Φ}^{2}-\frac{1}{2}{({∂}_{x}Φ)}^{2}

The Euler-Lagrange equation for field in terms of the Lagrangian density

ℒ(t,x)

, is given as,

\frac{∂ℒ}{{∂Φ}^{i}}={∂}_{μ}\left(\frac{∂ℒ}{∂({∂}_{μ}{Φ}^{i})}\right)

Substituting for

ℒ(t,x)

, we get Klein-Gordon equations in 1-dimension and 3-dimensions as,

\left({∂}_{t}^{2}-{∂}_{x}^{2}+𝓂^{2}\right)Φ=0

(1-D),

\left({∂}_{t}^{2}-{∂}_{x}^{2}-{∂}_{y}^{2}-{∂}_{z}^{2}+𝓂^{2}\right)Φ=0

Here the quantity

𝓂

represents, the interaction based mass density (mass for a unit area/volume), in the field

Φ(t,x)

    The objection to Kaluza's theory, was with his assumption of the existence of the 5th dimension, which was cylindrical.   For Kaluza, it was a mathematical construct which unified General Relativity with Electromagnetism. However if the 5th dimension exists then it must be observed experimentally.

    An attempt was made by Oscar Klein to introduce the quantum mechanics to explain the 5th dimension as follows:

Define a scalar field

Φ({x}^{M})=Φ({x}^{μ},{x}^{5})

↓

Write Klein-Gordon equation in 5-D as

\left({∂}_{t}^{2}-{∂}_{x}^{2}-{∂}_{y}^{2}-{∂}_{z}^{2}+{∂}_{5}^{2}+{𝓂}_0^{2}\right)Φ=0

↓

Introduce a boundary condition for the scalar field

Φ({x}^{M})

Φ({x}^{μ},{x}^{5})=Φ({x}^{μ},{x}^{5}+2πa)

a

is called compatification scale, assumed to be of the order of Planck's length)

↓

Assume a sinusoidal dependence for

Φ

on x⁵ and use harmonic expansion to obtain decoupled 4D Klein-Gordon Equation as,

\left({∂}_{t}^{2}-{∂}_{x}^{2}-{∂}_{y}^{2}-{∂}_{z}^{2}+{𝓂}_{n}^{2}\right)Φ({x}^{μ})=0

{𝓂}_{n}^2={𝓂}_{0}^2+\frac{{n}^{2}}{{a}^{2}}.

The problem immediately arose as the value of the mass of the 1^st mode m₁ would have been around 10¹⁹ GeV. The verification of such a high value of mass is not possible.

However, as we discussed earlier the mass

𝓂

in Klein-Gordon equation represents the interaction between oscillators, rather than the inertial mass of a particle (boson).

Perhaps the extrapolation of the classic 4-D Klein-Gordon equation to 5-D equation, based on the theory proposed by Kaluza, is not so straightforward. We simply can not go on adding space dimensions into a relativistic field equation beyond 3-D, while keeping

{∂}_{t}

, or

\widehat{{e}_{t}}

, unchanged.

The more fundamental issue is the independence of 4-D geometry from x⁵. At present, there is no satisfactory explanation for this independence.

Let us review the role of the metric tensor in geometric space. We have two descriptions, we can call bird's eye view of space-time (extrinsic view for the observer) and bug's eye view of space-time (intrinsic view of the observer in S-frame).

In General Relativity the space-time is curved (extrinsic view), however the observer measures the space as flat and time as linear (intrinsic view or S-frame). The metric tensor

{g}_{μν}

provides the relationship between both, so that the equations of motion can be derived in the flat-space.

We keep in mind that what we are calling extrinsic view or phase space , acts on every point (j-pixel) in the measurement space. Therefore the value of metric tensor may change from j-pixel to j-pixel in S-frame, for a curved phase space. And subsequently if we want a metric which is coupled to every j-pixel of space-time in GR, the metric must not change across j-pixels underlying the geometry in S-frame.

There is only one possibility as shown below, and it corresponds to Kaluza's cylindrical condition

\frac{∂}{{∂x}^{5}}=0

Therefore in a discrete measurement space, the structure formed by the measurement circuits in the abstract gauge space, must be cylindrical. Only then they can describe the geometry in the observer's frame of reference.

Furthermore since the structure represents the progression of measurements, it will have its own time axis

τ

along the path PQ, which is likely to be below Planck's scale. (Looks like we just contradicted quantum mechanics, which does not allow another intrinsic space-time description within S-frame, possibly because of the unitarity condition.)³

    We also note that since there is a displacement within this gauge space, it will result in establishing a 2D charge space, represented by M_ckt. The completed measurement in this charge space, i.e. completion of measurement for the path PQ, will result in the measurement of a unit of charge in the intrinsic space of the observer in S-frame.

    Additionally, a displacement in the phase space is likely to cause the physical motion in the intrinsic space of the observer in S-frame.

   Keeping in mind that

x

in phase space corresponds to its own time axis

τ

in phase space, the physical dimension

a

used in Kaluza-Klein theory and time

τ

both are well below the measurement capability of the observer Obs_1/137. Symbolically, we can write the scalar field as,

ϕ\left({x}^{μ},{x}^{5}\right)=ϕ\left({x}^{μ},{x}^{5}+2π{0}_{j}\right), \;a={0}_{j}.

The quantity

2π{0}_{j}

, is measurable in intrinsic phase space, but below the measurement threshold in S-frame.

The property of invariance under U(1) transformation

{e}^{iθx}

(Electromagnetic Field), is satisfied by each measurement circuit in the phase space.

{x}^{5} = {x}^{5}+θ(x),

ψ→{ψ}^{,}=U(x)ψ,

and\;U(x)={e}^{iθx}.

When we look for invariance of a wave-function under a gauge transformation, the space

x

refers to the intrinsic phase space in S-frame of the observer Obs_1/137. Thus the quantity

{ds}^{2}

remains unchanged with Kaluza ansatz.

The cylindrical structure proposed by Kaluza corresponds to M_ckt in discrete measurement space. It reflects the capacity of the maximum efficiency observer Obs_1/1 to make a measurement in a given field.

Since 4D-space is independent of variation in

{x}^{5}

, no internal structure for the cylindrical path PQ, can be measured in the flat-space or the intrinsic space of the macroscopic observer.

The methodology proposed by Klein, is an important one. We will be discussing its implications in the discrete measurement space next.

....to be continued

______________

1. If the observer travels the path PQ in one step, i.e. all the information contained within the path PQ is extracted by a single measurement, it is called zero-entropy measurement. The time-axis has no physical significance in this case as log_e1 = 0.

2. The Klein-Gordon equation describes the relativistic motion of composite particles with spin 0, or equivalently the measurements of a composite information

δ

-function in j-space. So the equation of motion of a Higgs boson can be described by Klein-Gordon equation.

3. Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton Series in Physics, 1997, page 68.


	The Theory of Measurements - VIB the 5^th? 4^th June 2025	HOME
“From my description of the goal one can perhaps see some of the difficulties. Since quantum gravity should be background-free, the geometrical structures defining the causal structure of spacetime should themselves be local degrees of freedom propagating causally. This much is already true in general relativity. But because quantum gravity should be a quantum theory, these degrees of freedom should be treated quantum-mechanically. So at the very least, we should develop a quantum theory of some sort of geometrical structure that can define a causal structure on spacetime.” - John C. Baez, Higher-Dimensional Algebra and Planck-Scale Physics, 1999. "Thus, the conclusion is that new ideas have arisen in loop quantum gravity which have some hope of resolving the problem of time and the problem of quantum cosmology. Further, as a well formulated background independent quantum theory, loop quantum gravity allows older ideas about these problems to be precisely formulated and tested. Meanwhile, while string theory apparently does not offer so far anything new to resolve these problems, it is striking that a few string theorists have put forward proposals that appear to be inspired by the ideas coming from loop quantum gravity. ” - Lee Smolin, How far are we from the quantum theory of gravity?, 2003. "This distinction between classical and quantum geometry is dramatically augmented by the realization that topology change — an operation which by fundamental definition in classical geometry is discontinuous — can be perfectly continuous and smooth in the quantum geometry of string theory. " - Brian R. Greene, String Theory on Calabi-Yau Manifolds, 1997.	We continue our discussion of the measurement of a path PQ being measured with a stack of measurement circuits. The path PQ represents an information $δ$ -function which is either singular or complex in nature. The information contained within the Path PQ, is extracted by the observer using M_ckts. If all the information is extracted, the observer has arrived at the point Q from the starting point P. The observer stays in the state of measurement until the measurement is complete.¹ It is important to note that the measurement circuits M_ckt, corresponds to the measurement capability of Obs_1/1 (Obs_c). The system in the state of the measurement i.e. the electron orbiting around a proton, corresponds to the measurement capability of the Obs_1/137 (Obs_elec). Equivalently, the vector space V corresponds to Obs_1/1, and the covector space V* corresponds to Obs_1/137, such that V:V $→$ $ℝ$ . We note that Lorentz Invariance / Möbius Invariance along with Locality and Unitarity, is implicit. All the physical measurements, are made and interpreted in S-frame. Let us quickly review the relativistic wave equation or Klein-Gordon equation first. We have a rather simple homemade criterion in j-space, based on the fact whether the measurement circuit M_ckt can be completed or not, while making a measurement. If the measurement circuit can be completed then we can write a relativistic wave equation for a spin-less particle, known as the Klein-Gordon equation. The idea is fairly straightforward. We set up a field based on oscillations (harmonic). We then write Lagrangian density of the field. Next, we use Euler-Lagrange equation to derive the equation of motion, which is the Klein-Gordon equation. An important point to note, is that mass appearing in this equation is based on interactions between oscillators. We provide a quick summary as follows. Consider a pair of coupled (interacting) harmonic oscillators as shown. We can write the Kinetic Energy T, Potential Energy V, and then the Lagrangian for the configuration as, $T = \frac{1}{2}m{\dot{{x}_{1}}}^{2}+\frac{1}{2}m{\dot{{x}_{2}}}^{2}$ , $V = \frac{1}{2} k x_{1}^{2} + \frac{1}{2} k x_{2}^{2} + \frac{1}{2} κ (x_{1} - x_{2})^{2}, and$ $L=T-V$ $=\frac{1}{2}m{\dot{{x}_{1}}}^{2}+\frac{1}{2}m{\dot{{x}_{2}}}^{2}-\frac{1}{2}k{x}_{1}^{2}-\frac{1}{2}k{x}_{2}^{2}-\frac{1}{2}κ({x}_{1}-{x}_{2})^{2}.$ Now, suppose we place these oscillators infinitesimally close together i.e. $\left({x}_{2}-{x}_{1}={0}_{j}\right)$ . ( The value of the 0_j is based on the measurement capability of the observer Obs_1/1, i.e. The locality condition. Therefore the precise measurement of 0_j is not possible for Obs_1/137, which has a lower measurement capability than Obs_1/1.) We select coordinate axes as, (i) x - the separation between the oscillators, and (ii) ϕ(x) - their respective vertical displacements also known as the field amplitudes. Imagine that we have infinite many of these oscillators infinitesimally close to each other. In this case, x is a continuous variable. And, rather than calculating total energy of the configuration, we write the Kinetic Energy Density $𝒯$ , Potential Energy Density $𝒱$ , and the Lagrangian density $ℒ$ , for the unit area of the configuration. $𝒯=\frac{1}{2}m{\dot{φ}}^{2}$ , and $𝒱=\frac{1}{2}k{φ}^{2}+\frac{1}{2}κ{{∂}_{x}φ}^{2}$ . Substitute $κ=m$ and $Φ=\sqrt{m}φ$ and then ${𝓂}^{2}=\frac{k}{m}$ to obtain, $𝒯=\frac{1}{2}({∂}_{t}Φ{)}^{2}$ , and $𝒱=\frac{1}{2}{𝓂}^{2}{Φ}^{2}+\frac{1}{2}{({∂}_{x}Φ)}^{2}$ . We note that by using the condition $κ=m$ , we have made the mass measured in the unit area, purely interaction dependent. We can write the Lagrangian density as, $ℒ(t,x)=𝒯(t,x)-𝒱(t,x)==\frac{1}{2}({∂}_{t}Φ{)}^{2}-\frac{1}{2}{𝓂}^{2}{Φ}^{2}-\frac{1}{2}{({∂}_{x}Φ)}^{2}$ . The Euler-Lagrange equation for field in terms of the Lagrangian density $ℒ(t,x)$ , is given as, $\frac{∂ℒ}{{∂Φ}^{i}}={∂}_{μ}\left(\frac{∂ℒ}{∂({∂}_{μ}{Φ}^{i})}\right)$ . Substituting for $ℒ(t,x)$ , we get Klein-Gordon equations in 1-dimension and 3-dimensions as, $\left({∂}_{t}^{2}-{∂}_{x}^{2}+𝓂^{2}\right)Φ=0$ (1-D), $\left({∂}_{t}^{2}-{∂}_{x}^{2}-{∂}_{y}^{2}-{∂}_{z}^{2}+𝓂^{2}\right)Φ=0$ Here the quantity $𝓂$ represents, the interaction based mass density (mass for a unit area/volume), in the field $Φ(t,x)$ . The objection to Kaluza's theory, was with his assumption of the existence of the 5th dimension, which was cylindrical. For Kaluza, it was a mathematical construct which unified General Relativity with Electromagnetism. However if the 5th dimension exists then it must be observed experimentally. An attempt was made by Oscar Klein to introduce the quantum mechanics to explain the 5th dimension as follows: Define a scalar field $Φ({x}^{M})=Φ({x}^{μ},{x}^{5})$ $↓$ Write Klein-Gordon equation in 5-D as $\left({∂}_{t}^{2}-{∂}_{x}^{2}-{∂}_{y}^{2}-{∂}_{z}^{2}+{∂}_{5}^{2}+{𝓂}_0^{2}\right)Φ=0$ . $↓$ Introduce a boundary condition for the scalar field $Φ({x}^{M})$ as $Φ({x}^{μ},{x}^{5})=Φ({x}^{μ},{x}^{5}+2πa)$ . $a$ is called compatification scale, assumed to be of the order of Planck's length) $↓$ Assume a sinusoidal dependence for $Φ$ on x⁵ and use harmonic expansion to obtain decoupled 4D Klein-Gordon Equation as, $\left({∂}_{t}^{2}-{∂}_{x}^{2}-{∂}_{y}^{2}-{∂}_{z}^{2}+{𝓂}_{n}^{2}\right)Φ({x}^{μ})=0$ , ${𝓂}_{n}^2={𝓂}_{0}^2+\frac{{n}^{2}}{{a}^{2}}.$ The problem immediately arose as the value of the mass of the 1^st mode m₁ would have been around 10¹⁹ GeV. The verification of such a high value of mass is not possible. However, as we discussed earlier the mass $𝓂$ in Klein-Gordon equation represents the interaction between oscillators, rather than the inertial mass of a particle (boson). Perhaps the extrapolation of the classic 4-D Klein-Gordon equation to 5-D equation, based on the theory proposed by Kaluza, is not so straightforward. We simply can not go on adding space dimensions into a relativistic field equation beyond 3-D, while keeping ${∂}_{t}$ , or $\widehat{{e}_{t}}$ , unchanged. The more fundamental issue is the independence of 4-D geometry from x⁵. At present, there is no satisfactory explanation for this independence. Let us review the role of the metric tensor in geometric space. We have two descriptions, we can call bird's eye view of space-time (extrinsic view for the observer) and bug's eye view of space-time (intrinsic view of the observer in S-frame). In General Relativity the space-time is curved (extrinsic view), however the observer measures the space as flat and time as linear (intrinsic view or S-frame). The metric tensor ${g}_{μν}$ provides the relationship between both, so that the equations of motion can be derived in the flat-space. We keep in mind that what we are calling extrinsic view or phase space , acts on every point (j-pixel) in the measurement space. Therefore the value of metric tensor may change from j-pixel to j-pixel in S-frame, for a curved phase space. And subsequently if we want a metric which is coupled to every j-pixel of space-time in GR, the metric must not change across j-pixels underlying the geometry in S-frame. There is only one possibility as shown below, and it corresponds to Kaluza's cylindrical condition $\frac{∂}{{∂x}^{5}}=0$ . Therefore in a discrete measurement space, the structure formed by the measurement circuits in the abstract gauge space, must be cylindrical. Only then they can describe the geometry in the observer's frame of reference. Furthermore since the structure represents the progression of measurements, it will have its own time axis $τ$ along the path PQ, which is likely to be below Planck's scale. (Looks like we just contradicted quantum mechanics, which does not allow another intrinsic space-time description within S-frame, possibly because of the unitarity condition.)³ We also note that since there is a displacement within this gauge space, it will result in establishing a 2D charge space, represented by M_ckt. The completed measurement in this charge space, i.e. completion of measurement for the path PQ, will result in the measurement of a unit of charge in the intrinsic space of the observer in S-frame. Additionally, a displacement in the phase space is likely to cause the physical motion in the intrinsic space of the observer in S-frame. Keeping in mind that $x$ in phase space corresponds to its own time axis $τ$ in phase space, the physical dimension $a$ used in Kaluza-Klein theory and time $τ$ both are well below the measurement capability of the observer Obs_1/137. Symbolically, we can write the scalar field as, $ϕ\left({x}^{μ},{x}^{5}\right)=ϕ\left({x}^{μ},{x}^{5}+2π{0}_{j}\right), \;a={0}_{j}.$ The quantity $2π{0}_{j}$ , is measurable in intrinsic phase space, but below the measurement threshold in S-frame. The property of invariance under U(1) transformation ${e}^{iθx}$ (Electromagnetic Field), is satisfied by each measurement circuit in the phase space. ${x}^{5} = {x}^{5}+θ(x),$ $ψ→{ψ}^{,}=U(x)ψ,$ $and\;U(x)={e}^{iθx}.$ When we look for invariance of a wave-function under a gauge transformation, the space $x$ refers to the intrinsic phase space in S-frame of the observer Obs_1/137. Thus the quantity ${ds}^{2}$ remains unchanged with Kaluza ansatz. The cylindrical structure proposed by Kaluza corresponds to M_ckt in discrete measurement space. It reflects the capacity of the maximum efficiency observer Obs_1/1 to make a measurement in a given field. Since 4D-space is independent of variation in ${x}^{5}$ , no internal structure for the cylindrical path PQ, can be measured in the flat-space or the intrinsic space of the macroscopic observer. The methodology proposed by Klein, is an important one. We will be discussing its implications in the discrete measurement space next. ....to be continued* ______________ 1. If the observer travels the path PQ in one step, i.e. all the information contained within the path PQ is extracted by a single measurement, it is called zero-entropy measurement. The time-axis has no physical significance in this case as log_e1 = 0. 2. The Klein-Gordon equation describes the relativistic motion of composite particles with spin 0, or equivalently the measurements of a composite information $δ$ -function in j-space. So the equation of motion of a Higgs boson can be described by Klein-Gordon equation. 3. Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton Series in Physics, 1997, page 68.	Previous Blogs: The Theory of Measurements - V IA The Theory of Measurements - V The Theory of Measurements - I V The Theory of Measurements - I I I The Theory of Measurements - I I The Theory of Measurements - I An Ecosystem of δ-Potentials - IVB An Ecosystem of δ-Potentials - IVA An Ecosystem of δ-Potentials - III An Ecosystem of δ-Potentials - II An Ecosystem of δ-Potentials - I Nutshell-2019 Stitching Measurement Space - III Stitching Measurement Space - II Stitching Measurement Space - I Mass Length & Topology A Timeless Constant Space Time and Entropy Nutshell-2018 Curve of Least Disorder Möbius & Lorentz Transformation - II Möbius & Lorentz Transformation - I Knots, DNA & Enzymes Quantum Comp - III Nutshell-2017 Quantum Comp - II Quantum Comp - I Insincere Symmetry - II Insincere Symmetry - I Existence in 3-D Infinite Source Nutshell-2016 Quanta-II Quanta-I EPR Paradox-II EPR Paradox-I De Broglie Equation Duality in j-space A Paradox The Observers Nutshell-2015 Chiral Symmetry Sigma-z and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous z-Axis
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