Theory of Measurements 6A

We are discussing a system in a state of measurement. We are particularly interested in the information structures while the state of measurement continues. We note, that no motion is possible for the system under consideration, until the measurement is complete.

As an example, we are evaluating a system consisting of an electron and a proton, while electron is in a state of measurement with respect to proton.

Both the electron and the proton are observers and they have their relativistic capabilities, defined by their respective fine structure constants. We can visualize the capabilities of various observers using the Light Cone, as shown here:

Here Obs_o (

α = 0

), represents the observer with infinite inertia or equivalently minimal measurement capability. Obs_1/137 represents an electron in a state of measurement. Similarly Obs_1/1 or Obs_c represents an observer with the maximum measurement capability (

α=1

), in the discrete measurement space. The proton, which is being measured by the electron, represents a structure with infinite information content which is equivalent to an observer with infinite capability(

α=\infty

).

The electron is assigned S-frame and the proton is assigned S'-frame. Obs_c sets up a field (

{F}^{μν}{F}_{μν}

in Lagrangian

ℒ

), without any delta function in it. The measurement configuration is shown below:

We will note that electron making measurements in S-frame or in

{F}^{μν}

field, has no awareness of the proton in S'-frame. All the physical measurements are made and interpreted in S-frame.

As an observer is making measurements, we have various scenarios:

Scenario 1. The observer in S-frame has enough resources, so that it can complete the measurement circuit M_ckt in S'-frame. There are two possibilities and they are shown below:

In first case, the normal vector

\widehat{n}

remains unchanged as it travels around the measurement circuit formed by the vectors u and v. This case represents the classical Galilean frame.

In second case, the direction of the normal vector

\widehat{n}

changes as it travels around a measurement circuit. In this case the measurement space has a curvature and the change in the direction of

\widehat{n}

, represents that.

This particular scenario represents, gravitational and electromagnetic phenomena which are deterministic in nature.

Scenario 2. We have another scenario, in which the measurement circuit can not be completed. We have to define a path PQ to complete the gap, as shown below:

In this case, the Riemann Curvature Tensor (RCT) characterizing the extrinsic measurement space, is defined as:

R(u,v)n={▽}_{u}{▽}_{v}n-{▽}_{v}{▽}_{u}n-{▽}_{[u,v]}n

The term

{▽}_{[u,v]}n

, closes the gap in the measurement circuit M_ckt. The measurement of the path PQ in the discrete measurement space, is where the Quantum Mechanics and the High Energy Physics come in while describing our observations.

As discussed previously, the bottom line in the discrete measurement space remains whether the observer in S-frame can complete the measurement circuit M_ckt in S'-frame, or not?

For now, we will be discussing the Scenario-1, where M_ckt can be completed in S'-frame. This is, because no motion is possible in the physical space without completing at least one measurement circuit, which represents the motion in abstract gauge space. (The Scenario-2 requires the use of spinors as a measurement tool. We will come to it in later blogs.)

We just introduced a rather vague phrase, "the motion in abstract gauge space". What does it really mean? Consider the following picture where a path PQ, is under measurement.

The path PQ is not in a conventional physical space. The path PQ represents an information

δ

-function in S'-frame, which may be simple or composite, being measured by the observer in S-frame. (We note that the

δ

-function is determined by the observer Obs₁. At the same time, neither Obs_1/137 nor Obs₁ can determine the true nature of proton.)

The path PQ exists in a measurement space extrinsic to the observer in S-frame, (for example,

{{J}_{D} =\frac{∂D}{∂t}}

, the Displacement Current Density for an electron)¹. The motion in the physical space intrinsic to the observer in S-frame, (Current Density J), can take place only when the measurement of the path PQ is completed, hence the term gauge space.

So while traveling along the path PQ, we found ourselves a cylinder in the gauge space (discrete measurement space or j-space), consisting of a stack of M_ckts. This is actually our way of counting in the measurement space. But then how do we connect it to what we already know, namely Einstein's theory of gravitation and Maxwell's theory of electromagnetism?

We will use the ansatz provided by Kaluza² discussed here. We will further discuss its validity in the discrete measurement space.

Note that, our objective is write a 5-dimensional metric tensor g_AB, which combines General Relativistic gravitational theory and Electromagnetic theories i.e.

g_AB

≡

GR metric tensor

{g}_{μν}⨂

Electromagnetic 4-potential

{A}_{μ}

↓

Define ${X}^{m}=\left({x}^{0},{x}^{1},{x}^{2},{x}^{3},{x}^{5}\right).$

$↓$

Write Kaluza metric as follows:

{g}_{AB}=\begin{pmatrix} {g}_{μν}+{φ}^{2}{A}_{μ}{A}_{ν} & {φ}^{2}{A}_{μ}\\ {φ}^{2}{A}_{ν} & {φ}^{2} \end{pmatrix}

↓

Place the cylindrical condition as shown:

\frac{∂}{∂{x}_{5}} = 0

,

or equivalently

{g}_{μν}

fields in S-frame, are independent of

\widehat{{e}_{5}}

, the basis vector for

{x}_{5}

. (

\widehat{{e}_{5}}

represents so-called 5^th dimension.)

↓

Calculate 5D Christoffel Symbols

{Γ}_{BL}^{A}{|}_{5D}

↓

Calculate 5D Riemann Curvature Tensor

{R}_{ALB}^{L}{|}_{5D}

↓

Calculate 5D Ricci Tensor

{R}_{AB}{|}_{5D}

↓

Calculate 5D Ricci Scalar

R

(By calculating 5D Ricci Scalar R, which is the contraction of 5D Ricci Tensor by Kaluza metric, we move from the extrinsic space to the observer's intrinsic space, i.e. "curved topology or abstract gauge space" to "flat local space or S-frame, loaded with ever increasing entropy".)

(5D Ricci Scalar is, what the observer Obs_1/137 looking for in S-frame which is the intrinsic space of the observer, to get the equations of motion. 4D Ricci Scalar leads to the source free gravitational theory. 5D Ricci Scalar combines 4D Ricci Scalar

R

to the covariant gradient of 4D Electromagnetic potential

{A}_{μ}

, or Electromagnetic Tensor

{F}_{μν}

{R|}_{5D} = {R|}_{4D}-\frac{1}{4}{F}_{μν}{F}^{μν}

. (S-frame)

(The scalar field

ϕ

does not exist in S-frame as the cylindrical condition

\frac{∂}{∂{x}^{5}} = 0

, is assumed.)

Also

{ℒ}_{5D}^{GR} = {ℒ}_{4D}^{GR}+{ℒ}_{4D}^{EM}.

(S-frame)

and \; {ds}^{2}{|}_{5D}={g}_{MN}{dx}^{M}{dx}^{N}.

Here ds² is not invariant under most of the transformations except U(1).

We note that, the entity M_ckt is just a concept we have been using to characterize the state of measurement. We are yet to connect the cylinder we constructed using M_ckt, in the discrete measurement space (j-space) or S-frame, to the physics as we know it. We will do it next.

....to be continued

______________

1. When we write the expression for the displacement current density as

{J}_{D}=\frac{∂\widebar{D}}{∂t}≡{∂}_{t}\widebar{D}≡\widehat{{e}^{t}}.({ϵ}_{0}\widebar{E}+\widebar{P}),

we introduce the basis-vector for the time coordinate, which sets up the time-axis for the epoch for the electron's state of measurement in S-frame.

2. Th. Kaluza, On the Unification Problem in Physics, K

\ddot{o}

nigsberg, December 22, 1921. The textbook, Einstein's Physics, by Ta-Pei Cheng, Oxford University Press, 2013, is an excellent reference on Kaluza' miracle. Another valuable resource is, The Dawning of Gauge Theory, Lochlainn O'Raifeartaigh, Princeton Series in Physics, 1997.


	The Theory of Measurements - VIA the 5^th? 4^th June 2025	HOME
“The intellect seeking after an integrated theory cannot rest content with the assumption that there exist two distinct fields totally independent of each other by their nature,” - Albert Einstein, Noble Lecture 1923. "The metric is not a property of the world [space-time] in itself, rather space-time as a form of appearance is a completely formless four-dimensional continuum in the sense of analysis situs, but the metric expresses something real, something which exists in the world, which exerts through centrifugal and gravitational forces physical effects on matter, and whose state is conversely conditioned through the distribution and nature of matter. ” - Herman Weyl, as quoted in Stanford Encyclopedia of Philosophy. "The dualistic nature of gravitation and electricity still remaining here does not actually destroy the ensnaring beauty of either theory but rather affords a new challenge towards their triumph through an entirely unified picture of the world." - Th. Kaluza, On the Unification Problem in Physics.	We are discussing a system in a state of measurement. We are particularly interested in the information structures while the state of measurement continues. We note, that no motion is possible for the system under consideration, until the measurement is complete. As an example, we are evaluating a system consisting of an electron and a proton, while electron is in a state of measurement with respect to proton. Both the electron and the proton are observers and they have their relativistic capabilities, defined by their respective fine structure constants. We can visualize the capabilities of various observers using the Light Cone, as shown here: Here Obs_o ( $α = 0$ ), represents the observer with infinite inertia or equivalently minimal measurement capability. Obs_1/137 represents an electron in a state of measurement. Similarly Obs_1/1 or Obs_c represents an observer with the maximum measurement capability ( $α=1$ ), in the discrete measurement space. The proton, which is being measured by the electron, represents a structure with infinite information content which is equivalent to an observer with infinite capability( $α=\infty$ ). The electron is assigned S-frame and the proton is assigned S'-frame. Obs_c sets up a field ( ${F}^{μν}{F}_{μν}$ in Lagrangian $ℒ$ ), without any delta function in it. The measurement configuration is shown below: We will note that electron making measurements in S-frame or in ${F}^{μν}$ field, has no awareness of the proton in S'-frame. All the physical measurements are made and interpreted in S-frame. As an observer is making measurements, we have various scenarios: Scenario 1. The observer in S-frame has enough resources, so that it can complete the measurement circuit M_ckt in S'-frame. There are two possibilities and they are shown below: In first case, the normal vector $\widehat{n}$ remains unchanged as it travels around the measurement circuit formed by the vectors u and v. This case represents the classical Galilean frame. In second case, the direction of the normal vector $\widehat{n}$ changes as it travels around a measurement circuit. In this case the measurement space has a curvature and the change in the direction of $\widehat{n}$ , represents that. This particular scenario represents, gravitational and electromagnetic phenomena which are deterministic in nature. Scenario 2. We have another scenario, in which the measurement circuit can not be completed. We have to define a path PQ to complete the gap, as shown below: In this case, the Riemann Curvature Tensor (RCT) characterizing the extrinsic measurement space, is defined as: $R(u,v)n={▽}_{u}{▽}_{v}n-{▽}_{v}{▽}_{u}n-{▽}_{[u,v]}n$ . The term ${▽}_{[u,v]}n$ , closes the gap in the measurement circuit M_ckt. The measurement of the path PQ in the discrete measurement space, is where the Quantum Mechanics and the High Energy Physics come in while describing our observations. As discussed previously, the bottom line in the discrete measurement space remains whether the observer in S-frame can complete the measurement circuit M_ckt in S'-frame, or not? For now, we will be discussing the Scenario-1, where M_ckt can be completed in S'-frame. This is, because no motion is possible in the physical space without completing at least one measurement circuit, which represents the motion in abstract gauge space. (The Scenario-2 requires the use of spinors as a measurement tool. We will come to it in later blogs.) We just introduced a rather vague phrase, "the motion in abstract gauge space". What does it really mean? Consider the following picture where a path PQ, is under measurement. The path PQ is not in a conventional physical space. The path PQ represents an information $δ$ -function in S'-frame, which may be simple or composite, being measured by the observer in S-frame. (We note that the $δ$ -function is determined by the observer Obs₁. At the same time, neither Obs_1/137 nor Obs₁ can determine the true nature of proton.) The path PQ exists in a measurement space extrinsic to the observer in S-frame, (for example, ${{J}_{D} =\frac{∂D}{∂t}}$ , the Displacement Current Density for an electron)¹. The motion in the physical space intrinsic to the observer in S-frame, (Current Density J), can take place only when the measurement of the path PQ is completed, hence the term gauge space. So while traveling along the path PQ, we found ourselves a cylinder in the gauge space (discrete measurement space or j-space), consisting of a stack of M_ckts. This is actually our way of counting in the measurement space. But then how do we connect it to what we already know, namely Einstein's theory of gravitation and Maxwell's theory of electromagnetism? We will use the ansatz provided by Kaluza² discussed here. We will further discuss its validity in the discrete measurement space. Note that, our objective is write a 5-dimensional metric tensor g_AB, which combines General Relativistic gravitational theory and Electromagnetic theories i.e. g_AB $≡$ GR metric tensor ${g}_{μν}⨂$ Electromagnetic 4-potential ${A}_{μ}$ . $↓$ Define ${X}^{m}=\left({x}^{0},{x}^{1},{x}^{2},{x}^{3},{x}^{5}\right).$ $↓$ Write Kaluza metric as follows: ${g}_{AB}=\begin{pmatrix} {g}_{μν}+{φ}^{2}{A}_{μ}{A}_{ν} & {φ}^{2}{A}_{μ}\\ {φ}^{2}{A}_{ν} & {φ}^{2} \end{pmatrix}$ $↓$ Place the cylindrical condition as shown: $\frac{∂}{∂{x}_{5}} = 0$ , or equivalently ${g}_{μν}$ fields in S-frame, are independent of $\widehat{{e}_{5}}$ , the basis vector for ${x}_{5}$ . ( $\widehat{{e}_{5}}$ represents so-called 5^th dimension.) $↓$ Calculate 5D Christoffel Symbols ${Γ}_{BL}^{A}{\|}_{5D}$ $↓$ Calculate 5D Riemann Curvature Tensor ${R}_{ALB}^{L}{\|}_{5D}$ $↓$ Calculate 5D Ricci Tensor ${R}_{AB}{\|}_{5D}$ $↓$ Calculate 5D Ricci Scalar $R$ (By calculating 5D Ricci Scalar R, which is the contraction of 5D Ricci Tensor by Kaluza metric, we move from the extrinsic space to the observer's intrinsic space, i.e. "curved topology or abstract gauge space" to "flat local space or S-frame, loaded with ever increasing entropy".) (5D Ricci Scalar is, what the observer Obs_1/137 looking for in S-frame which is the intrinsic space of the observer, to get the equations of motion. 4D Ricci Scalar leads to the source free gravitational theory. 5D Ricci Scalar combines 4D Ricci Scalar $R$ to the covariant gradient of 4D Electromagnetic potential ${A}_{μ}$ , or Electromagnetic Tensor ${F}_{μν}$ .) ${R\|}_{5D} = {R\|}_{4D}-\frac{1}{4}{F}_{μν}{F}^{μν}$ . (S-frame) (The scalar field $ϕ$ does not exist in S-frame as the cylindrical condition $\frac{∂}{∂{x}^{5}} = 0$ , is assumed.) Also ${ℒ}_{5D}^{GR} = {ℒ}_{4D}^{GR}+{ℒ}_{4D}^{EM}.$ (S-frame) $and \; {ds}^{2}{\|}_{5D}={g}_{MN}{dx}^{M}{dx}^{N}.$ Here ds² is not invariant under most of the transformations except U(1). We note that, the entity M_ckt is just a concept we have been using to characterize the state of measurement. We are yet to connect the cylinder we constructed using M_ckt, in the discrete measurement space (j-space) or S-frame, to the physics as we know it. We will do it next. ....to be continued ______________ 1. When we write the expression for the displacement current density as ${J}_{D}=\frac{∂\widebar{D}}{∂t}≡{∂}_{t}\widebar{D}≡\widehat{{e}^{t}}.({ϵ}_{0}\widebar{E}+\widebar{P}),$ we introduce the basis-vector for the time coordinate, which sets up the time-axis for the epoch for the electron's state of measurement in S-frame. 2. Th. Kaluza, On the Unification Problem in Physics, K $\ddot{o}$ nigsberg, December 22, 1921. The textbook, Einstein's Physics, by Ta-Pei Cheng, Oxford University Press, 2013, is an excellent reference on Kaluza' miracle. Another valuable resource is, The Dawning of Gauge Theory, Lochlainn O'Raifeartaigh, Princeton Series in Physics, 1997.	Previous Blogs: 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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