**Relational Quantum Mechanics**

**Relational Quantum Mechanics**

A. Nicolaidis Additional information is available at the end of the chapter

A. Nicolaidis

26 Quantum Mechanics

360 Advances in Quantum Mechanics

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54892

### **1. Introduction**

Quantum mechanics (QM) stands out as the theory of the 20th century, shaping the most diverse phenomena, from subatomic physics to cosmology. All quantum predictions have been crowned with full success and utmost accuracy. Yet, the admiration we feel towards QM is mixed with surprise and uneasiness. QM defies common sense and common logic. Various paradoxes, including Schrodinger's cat and EPR paradox, exemplify the lurking conflict. The reality of the problem is confirmed by the Bell's inequalities and the GHZ equalities. We are thus led to revisit a number of old interlocked oppositions: operator – operand, discrete – continuous, finite –infinite, hardware – software, local – global, particular – universal, syntax – semantics, ontological – epistemological.

The logic of a physical theory reflects the structure of the propositions describing the physical system under study. The propositional logic of classical mechanics is Boolean logic, which is based on set theory. A set theory is deprived of any structure, being a plurality of structure-less individuals, qualified only by membership (or non-membership). Accordingly a set-theoretic enterprise is analytic, atomistic, arithmetic. It was noticed as early as 1936 by Neumann and Birkhoff that the quantum real needs a non-Boolean logical structure. On numerous cases the need for a novel system of logical syntax is evident. Quantum measurement bypasses the old disjunctions subject-object, observer-observed. The observer affects the system under observation and the borderline between ontological and epistemological is blurred. Correlations are not anymore local and a quantum system embodies multiple entanglements. The particular-universal dichotomy is also under revision. While a single quantum event is particular, a plethora of quantum events leads to universal patterns. Viewing the quantum system as a system encoding information, we understand that the usual distinction between hardware and software is not relevant. Most importantly, if we consider the opposing terms being-becoming, we realize that the emphasis is sifted to the becoming, the movement, the process. The underlying dynamics is governed by relational

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Nicolaidis; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Nicolaidis; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Nicolaidis, licensee InTech. This is an open access chapter distributed under the terms of the Creative

principles and we have suggested [1] that the relational logic of C. S. Peirce may serve as the conceptual foundation of QM.

Peirce, the founder of American pragmatism, made important contributions in science, philosophy, semeiotics and notably in logic. Many scholars (Clifford, Schröder, Whitehead, Lukasiewicz) rank Peirce with Leibniz and Aristotle in the history of thought. Logic, in its most general sense, is the formal science of representation, coextensive with semeiotics. Algebraic logic attempts to express the laws of thought in the form of mathematical equations, and Peirce incorporated a theory of relations into algebraic logic [2, 3]. Relation is the primary irreducible datum and everything is expressed in terms of relations. A relational formulation is bound to be synthetic, holistic, geometric. Peirce invented also a notation for quantifiers and developed quantification theory, thus he is regarded as one of the principal founders of modern logic.

In the next section we present the structures of the relational logic and a representation of relation which will lead us to the probability rule of QM. In the third section we analyze a discrete system and demonstrate the non-commutation of conjugate operators. In the last section we present the conclusions and indicate directions for future work.

#### **2. The logic of relations and the quantum rules**

The starting point is the binary relation *SiRS<sup>j</sup>* between the two 'individual terms' (subjects) *S<sup>j</sup>* and *Si*. In a short hand notation we represent this relation by *Rij* . Relations may be composed: whenever we have relations of the form *Rij* , *Rjl*, a third transitive relation *Ril* emerges following the rule [2, 3]

$$R\_{ij}R\_{kl} = \delta\_{jk}R\_{il} \tag{1}$$

the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category [5] consists of objects A, B, C,... and arrows (morphisms) f, g, h,... . Each arrow f is assigned an object *A* as domain and an object *B* as codomain, indicated by writing *<sup>f</sup>* : *<sup>A</sup>* <sup>→</sup> *<sup>B</sup>*. If *<sup>g</sup>* is an arrow *<sup>g</sup>* : *<sup>B</sup>* <sup>→</sup> *<sup>C</sup>* with domain *<sup>B</sup>*, the codomain of f, then f and g can be "composed" to give an arrow *gof* : *<sup>A</sup>* <sup>→</sup> *<sup>C</sup>*. The composition obeys the associative law *ho*(*gof*)=(*hog*)*of*. For each object *A* there is an arrow 1*<sup>A</sup>* : *<sup>A</sup>* <sup>→</sup> *<sup>A</sup>* called the identity arrow of *<sup>A</sup>*. The analogy with the relational logic of Peirce is evident, *Rij* stands as an arrow, the composition rule is manifested in eq. (1) and the identity arrow for *A* ≡ *S<sup>i</sup>* is *Rii*. There is an important literature on possible ways the category notions can be applied to physics; specifically to quantising space-time [6], attaching a formal language to a physical system [7], studying topological quantum field

A relation *Rij* may receive multiple interpretations: as the proof of the logical proposition *i* starting from the logical premise *j*, as a transition from the *j* state to the *i* state, as a measurement process that rejects all impinging systems except those in the state *j* and permits only systems in the state *i* to emerge from the apparatus. We proceed to a

*Rij* = |*ri*�

where state �*ri*| is the dual of the state|*ri*� and they obey the orthonormal condition

�*ri*| *r<sup>j</sup>* 

 *i*

<sup>|</sup>*̺i*� <sup>=</sup> <sup>1</sup> <sup>√</sup>*<sup>N</sup>*

All relations remain satisfied if we replace the state |*ri*� by |*̺i*�, where

It is immediately seen that our representation satisfies the composition rule eq. (1). The

 *n*

with *N* the number of states. Thus we verify Peirce's suggestion, eq. (2), and the state |*ri*� is derived as the sum of all its interactions with the other states. *Rij* acts as a projection,

We may think also of another property characterizing our states and define a corresponding

 *qj* 

*Qij* = |*qi*�

 *rj* 

(6)

Relational Quantum Mechanics http://dx.doi.org/10.5772/54892 363

= *δij* (7)

<sup>|</sup>*ri*� �*ri*<sup>|</sup> <sup>=</sup> **<sup>1</sup>** (8)

*Rij* |*rk*� = *δjk* |*ri*� . (10)


(11)

theories [8, 9], exploring quantum issues and quantum information theory [10].

representation of *Rij*

operator

completeness, eq.(5), takes the form

transferring from one *r* state to another *r* state

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce, in an original move, considered the individual as the aggregate of all its relations

$$S\_i = \sum\_j R\_{ij}.\tag{2}$$

It is easy to verify that the individual *S<sup>i</sup>* thus defined is an eigenstate of the *Rii* relation

$$R\_{ii} S\_i = S\_i. \tag{3}$$

The relations *Rii* are idempotent

$$R\_{ii}^2 = R\_{ii} \tag{4}$$

and they span the identity

$$\sum\_{i} R\_{ii} = \mathbf{1} \tag{5}$$

The Peircean logical structure bears great resemblance to category theory, a remarkably rich branch of mathematics developed by Eilenberg and Maclane in 1945 [4]. In categories the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category [5] consists of objects A, B, C,... and arrows (morphisms) f, g, h,... . Each arrow f is assigned an object *A* as domain and an object *B* as codomain, indicated by writing *<sup>f</sup>* : *<sup>A</sup>* <sup>→</sup> *<sup>B</sup>*. If *<sup>g</sup>* is an arrow *<sup>g</sup>* : *<sup>B</sup>* <sup>→</sup> *<sup>C</sup>* with domain *<sup>B</sup>*, the codomain of f, then f and g can be "composed" to give an arrow *gof* : *<sup>A</sup>* <sup>→</sup> *<sup>C</sup>*. The composition obeys the associative law *ho*(*gof*)=(*hog*)*of*. For each object *A* there is an arrow 1*<sup>A</sup>* : *<sup>A</sup>* <sup>→</sup> *<sup>A</sup>* called the identity arrow of *<sup>A</sup>*. The analogy with the relational logic of Peirce is evident, *Rij* stands as an arrow, the composition rule is manifested in eq. (1) and the identity arrow for *A* ≡ *S<sup>i</sup>* is *Rii*. There is an important literature on possible ways the category notions can be applied to physics; specifically to quantising space-time [6], attaching a formal language to a physical system [7], studying topological quantum field theories [8, 9], exploring quantum issues and quantum information theory [10].

2 Quantum Mechanics

the conceptual foundation of QM.

founders of modern logic.

emerges following the rule [2, 3]

The relations *Rii* are idempotent

and they span the identity

relations

principles and we have suggested [1] that the relational logic of C. S. Peirce may serve as

Peirce, the founder of American pragmatism, made important contributions in science, philosophy, semeiotics and notably in logic. Many scholars (Clifford, Schröder, Whitehead, Lukasiewicz) rank Peirce with Leibniz and Aristotle in the history of thought. Logic, in its most general sense, is the formal science of representation, coextensive with semeiotics. Algebraic logic attempts to express the laws of thought in the form of mathematical equations, and Peirce incorporated a theory of relations into algebraic logic [2, 3]. Relation is the primary irreducible datum and everything is expressed in terms of relations. A relational formulation is bound to be synthetic, holistic, geometric. Peirce invented also a notation for quantifiers and developed quantification theory, thus he is regarded as one of the principal

In the next section we present the structures of the relational logic and a representation of relation which will lead us to the probability rule of QM. In the third section we analyze a discrete system and demonstrate the non-commutation of conjugate operators. In the last

The starting point is the binary relation *SiRS<sup>j</sup>* between the two 'individual terms' (subjects) *S<sup>j</sup>* and *Si*. In a short hand notation we represent this relation by *Rij* . Relations may be composed: whenever we have relations of the form *Rij* , *Rjl*, a third transitive relation *Ril*

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce, in an original move, considered the individual as the aggregate of all its

> *S<sup>i</sup>* = *j*

It is easy to verify that the individual *S<sup>i</sup>* thus defined is an eigenstate of the *Rii* relation

*R*2

 *i*

The Peircean logical structure bears great resemblance to category theory, a remarkably rich branch of mathematics developed by Eilenberg and Maclane in 1945 [4]. In categories

*RijRkl* = *jkRil* (1)

*Rij* . (2)

*RiiS<sup>i</sup>* = *Si*. (3)

*ii* = *Rii* (4)

*Rii* = **1** (5)

section we present the conclusions and indicate directions for future work.

**2. The logic of relations and the quantum rules**

A relation *Rij* may receive multiple interpretations: as the proof of the logical proposition *i* starting from the logical premise *j*, as a transition from the *j* state to the *i* state, as a measurement process that rejects all impinging systems except those in the state *j* and permits only systems in the state *i* to emerge from the apparatus. We proceed to a representation of *Rij*

$$R\_{ij} = |r\_i\rangle\left\langle r\_j \right| \tag{6}$$

where state �*ri*| is the dual of the state|*ri*� and they obey the orthonormal condition

$$
\left< r\_i \right| r\_j \right> = \delta\_{ij} \tag{7}
$$

It is immediately seen that our representation satisfies the composition rule eq. (1). The completeness, eq.(5), takes the form

$$\sum\_{i} |r\_{i}\rangle\langle r\_{i}| = \mathbf{1} \tag{8}$$

All relations remain satisfied if we replace the state |*ri*� by |*̺i*�, where

$$\left| \varrho\_i \right\rangle = \frac{1}{\sqrt{N}} \sum\_n \left| r\_i \right\rangle \left\langle r\_n \right| \tag{9}$$

with *N* the number of states. Thus we verify Peirce's suggestion, eq. (2), and the state |*ri*� is derived as the sum of all its interactions with the other states. *Rij* acts as a projection, transferring from one *r* state to another *r* state

$$R\_{ij} \left| r\_k \right> = \delta\_{jk} \left| r\_i \right> . \tag{10}$$

We may think also of another property characterizing our states and define a corresponding operator

$$Q\_{ij} = |q\_i\rangle\langle q\_j|\tag{11}$$

with

$$Q\_{ij} \left| q\_k \right> = \delta\_{jk} \left| q\_i \right> \tag{12}$$

The completeness relation, eq. (13), guarantees that *p* (*q<sup>j</sup>* , *ri*) may assume the role of a

We discover that starting from the relational logic of Peirce we obtain the essential law of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic

Consider a chain of *N* discrete states |*ak*�, with *k* = 1, 2, *...* , *N*. A relation *R* acts like a

*p* (*q<sup>j</sup>* , *ri*) = 1. (23)

Relational Quantum Mechanics http://dx.doi.org/10.5772/54892 365

*R* |*ak*� = |*ak*+1� (24) *R* |*a<sup>N</sup>* � = |*a*1� (25)

*R<sup>N</sup>* = **1** (26)

*j*=0

 *R ak*

*R* |*bi*� = *b<sup>i</sup>* |*bi*� (29)

(30)

*j*

= 0 (28)

(31)

*k* = 1, 2, *...* , *N* (27)

 *j*

probability since

shift operator

*N* is the period of *R*

Then we have

measurement presented by Schwinger [11].

**3. The emergence of Planck's constant**

The numbers which satisfy *a<sup>N</sup>* = 1 are given by

*<sup>R</sup><sup>N</sup>* <sup>−</sup> <sup>1</sup> <sup>=</sup>

*R* has a set of eigenfunctions

Notice that we may write

*<sup>a</sup><sup>k</sup>* <sup>=</sup> exp

*<sup>N</sup>*

 *R ak*

with *b<sup>i</sup>* the *N*-th root of unity (*b<sup>i</sup>* = *ai*). It is decomposed like

 *bj b<sup>j</sup>* <sup>=</sup> <sup>1</sup> *N N*

<sup>2</sup>*πi <sup>k</sup> N* 

− 1 =

*R* = *j bj bj b<sup>j</sup>* 

 *R ak* − 1 *N* −1

*k*=1

 *R bj* *k*

and

$$\sum\_{i} |q\_{i}\rangle\langle q\_{i}| = \mathbf{1}.\tag{13}$$

Successive measurements of the *q*-ness and *r*-ness of the states is provided by the operator

$$R\_{ij}Q\_{kl} = |r\_i\rangle\left\langle r\_j \right| \left. q\_k \right\rangle\left\langle q\_l \right| = \left\langle r\_j \right| \left. q\_k \right\rangle S\_{il} \tag{14}$$

with

$$S\_{il} = \left| r\_i \right\rangle \left\langle q\_l \right|. \tag{15}$$

Considering the matrix elements of an operator *A* as *Anm* = �*r<sup>n</sup>* |*A*| *rm*� we find for the trace

$$Tr\left(S\_{il}\right) = \sum\_{n} \left< r\_n \left| S\_{il} \right| \left| r\_n \right> = \left< q\_l \right| \left| r\_i \right> . \tag{16}$$

From the above relation we deduce

$$Tr\left(R\_{ij}\right) = \delta\_{ij}.\tag{17}$$

Any operator can be expressed as a linear superposition of the *Rij*

$$A = \sum\_{i,j} A\_{ij} R\_{ij} \tag{18}$$

with

$$A\_{ij} = \operatorname{Tr}\left(AR\_{ji}\right). \tag{19}$$

The individual states can be redefined

$$|r\_i\rangle \to e^{i\varphi\_i} |r\_i\rangle \tag{20}$$

$$|q\_i\rangle \to e^{i\theta\_i} |q\_i\rangle \tag{21}$$

without affecting the corresponding composition laws. However the overlap number �*ri*| *q<sup>j</sup>* changes and therefore we need an invariant formulation for the transition <sup>|</sup>*ri*� → *qj* . This is provided by the trace of the closed operation *RiiQjjRii*

$$\operatorname{Tr}\left(R\_{ii}Q\_{jj}R\_{ii}\right) \equiv p\left(q\_j, r\_i\right) = \left|\left\right|^2. \tag{22}$$

The completeness relation, eq. (13), guarantees that *p* (*q<sup>j</sup>* , *ri*) may assume the role of a probability since

$$\sum\_{j} p\left(q\_{j}, r\_{i}\right) = 1.\tag{23}$$

We discover that starting from the relational logic of Peirce we obtain the essential law of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic measurement presented by Schwinger [11].

#### **3. The emergence of Planck's constant**

Consider a chain of *N* discrete states |*ak*�, with *k* = 1, 2, *...* , *N*. A relation *R* acts like a shift operator

$$R\left|a\_{k}\right\rangle = \left|a\_{k+1}\right\rangle\tag{24}$$

$$R|a\_N\rangle = |a\_1\rangle\tag{25}$$

*N* is the period of *R*

4 Quantum Mechanics

*Qij* |*qk*� = *δjk* |*qi*� (12)

<sup>|</sup>*qi*� �*qi*<sup>|</sup> <sup>=</sup> **<sup>1</sup>**. (13)

*Sil* = |*ri*� �*ql*| . (15)

�*r<sup>n</sup>* |*Sil*| *rn*� = �*ql*| *ri*� . (16)

*T r* (*Rij* ) = *δij* . (17)

*Aij* = *T r* (*ARji*). (19)

�*ri*<sup>|</sup> *<sup>q</sup><sup>j</sup>* 

*AijRij* (18)

*iϕ<sup>i</sup>* <sup>|</sup>*ri*� (20)

*iθ<sup>i</sup>* <sup>|</sup>*qi*� (21)

*qj* . This

<sup>2</sup> . (22)

*<sup>q</sup>k*� *<sup>S</sup>il* (14)

 *i*

> *rj*

*RijQkl* = |*ri*�

*T r* (*Sil*) =

Any operator can be expressed as a linear superposition of the *Rij*

*n*

*A* = *i*,*j*



changes and therefore we need an invariant formulation for the transition <sup>|</sup>*ri*� →

*T r* (*RiiQjjRii*) <sup>≡</sup> *<sup>p</sup>* (*q<sup>j</sup>* , *<sup>r</sup>i*) <sup>=</sup>

is provided by the trace of the closed operation *RiiQjjRii*

without affecting the corresponding composition laws. However the overlap number �*ri*| *q<sup>j</sup>*

Successive measurements of the *q*-ness and *r*-ness of the states is provided by the operator

Considering the matrix elements of an operator *A* as *Anm* = �*r<sup>n</sup>* |*A*| *rm*� we find for the

*<sup>q</sup>k*� �*ql*<sup>|</sup> <sup>=</sup>

*rj* 

with

and

with

trace

with

From the above relation we deduce

The individual states can be redefined

$$R^N = \mathbf{1} \tag{26}$$

The numbers which satisfy *a<sup>N</sup>* = 1 are given by

$$a\_k = \exp\left(2\pi i \frac{k}{N}\right) \quad k = 1, 2, \ldots, N \tag{27}$$

Then we have

$$R^N - 1 = \left(\frac{R}{a\_k}\right)^N - 1 = \left[\left(\frac{R}{a\_k}\right) - 1\right] \sum\_{j=0}^{N-1} \left(\frac{R}{a\_k}\right)^j = 0\tag{28}$$

*R* has a set of eigenfunctions

$$R\left|b\_{i}\right\rangle = b\_{i}\left|b\_{i}\right\rangle\tag{29}$$

with *b<sup>i</sup>* the *N*-th root of unity (*b<sup>i</sup>* = *ai*). It is decomposed like

$$R = \sum\_{j} b\_{j} \left| b\_{j} \right\rangle \left\langle b\_{j} \right| \tag{30}$$

Notice that we may write

$$\left| b\_{j} \right\rangle \left\langle b\_{j} \right| = \frac{1}{N} \sum\_{k=1}^{N} \left( \frac{R}{b\_{j}} \right)^{k} \tag{31}$$

The above projection operator acting upon |*a<sup>N</sup>* � will give

$$\left| \left| b\_{j} \right> \left< b\_{j} \right| a\_{N} \right> = \frac{1}{N} \sum\_{k=1}^{N} \left( \frac{1}{b\_{j}} \right)^{k} \left| a\_{k} \right> \tag{32}$$

We conclude that the conjugate operators *R* and *Q* do not commute

Similarly

Then

(48), we deduce

the Compton wavelength.

*QR* <sup>=</sup> exp

*<sup>Q</sup>nR<sup>m</sup>* <sup>=</sup> exp

exp

unit measuring the number of phase space cells. Using rather exp *<sup>i</sup>*

<sup>2</sup>*πi* <sup>1</sup> *N* 

*QR* <sup>=</sup> exp *<sup>i</sup>*

Approaching the continuum we may replace the discrete operators by exponential forms

*<sup>R</sup>* <sup>=</sup> exp *<sup>i</sup>*

*<sup>Q</sup>* <sup>=</sup> exp *<sup>i</sup>*

With *R* and *Q* unitary operators, *X* and *P* are hermitian operators. From equs. (46), (47),

The foundational non-commutative law of Quantum Mechanics testifies that there is a limit size ¯*h* ∼ *pa* in dividing the phase space. With *p* ∼ *mv* ≃ *mc* we understand that *a* represents

and many options are available. Let us define

<sup>2</sup>*πi* <sup>1</sup> *N* 

<sup>2</sup>*πinm N* 

In our discrete model the non-commutativity is determined by *N*. As *N* → ∞ the relation-operators *Q* and *R* commute. However it would be hasty to conclude that as *N* → ∞ we reach the continuum. The transition from the discrete to the continuum is a subtle affair

*<sup>L</sup>* <sup>=</sup> *Na p* <sup>=</sup> <sup>2</sup>*<sup>π</sup>*

What counts is the size of the available phase space and we may use Planck's constant *h* as a

¯*hpa*

¯*hpX*

¯*haP*

*RQ* (42)

Relational Quantum Mechanics http://dx.doi.org/10.5772/54892 367

*<sup>R</sup>mQ<sup>n</sup>* (43)

*<sup>L</sup>* (44)

= exp [*ipa*] . (45)

¯*<sup>h</sup> pa*

*RQ* (46)

. (48)

[*X*, *P*] = *i*¯*h*. (49)

, equ.(42) becomes

(47)

Matching from the right with �*a<sup>N</sup>* | we obtain

$$
\left\langle a\_N \right| b\_j \right\rangle \left\langle b\_j \right| a\_N \rangle = \frac{1}{N} \tag{33}
$$

We adopt the positive root

$$
\left\langle b\_j \right| a\_N \rangle = \frac{1}{\sqrt{N}} \tag{34}
$$

and equ. (32) becomes

$$\left| b\_{j} \right> = \frac{1}{\sqrt{N}} \sum\_{k=1}^{N} \exp \left[ -2\pi i \frac{jk}{N} \right] \left| a\_{k} \right> \tag{35}$$

Inversely we have the decomposition

$$\left| \left| a\_{m} \right\rangle = \frac{1}{\sqrt{N}} \sum\_{n=1}^{N} \exp\left[2\pi i \frac{mn}{N}\right] \left| a\_{n} \right\rangle. \tag{36}$$

We introduce another relation *Q* acting like shift operator

$$
\langle b\_k | Q = \langle b\_{k+1} | \tag{37}
$$

$$
\left\langle b\_N \right| Q = \left\langle b\_1 \right| \tag{38}
$$

The relation *Q* receives the decomposition

$$Q = \sum\_{j} a\_{j} \left| a\_{j} \right\rangle \left\langle a\_{j} \right| \tag{39}$$

Consider now

$$\left\langle b\_{k} \right| QR = \left\langle b\_{k+1} \right| R = \exp\left[2\pi i \frac{(k+1)}{N}\right] \left\langle b\_{k+1} \right| \tag{40}$$

$$\left\langle b\_{k} \right| RQ = \exp\left[2\pi i \frac{k}{N}\right] \left\langle b\_{k} \right| Q = \exp\left[2\pi i \frac{k}{N}\right] \left\langle b\_{k+1} \right| \tag{41}$$

We conclude that the conjugate operators *R* and *Q* do not commute

$$QR = \exp\left[2\pi i \frac{1}{N}\right] RQ\tag{42}$$

Similarly

6 Quantum Mechanics

The above projection operator acting upon |*a<sup>N</sup>* � will give

 *bj b<sup>j</sup>* 

 *bj* <sup>=</sup> <sup>1</sup> <sup>√</sup>*<sup>N</sup>*

We introduce another relation *Q* acting like shift operator

<sup>|</sup>*am*� <sup>=</sup> <sup>1</sup>

<sup>√</sup>*<sup>N</sup>*

*Q* = *j aj aj a<sup>j</sup>* 

�*bk*| *QR* = �*bk*+1| *R* = exp

 <sup>2</sup>*πi <sup>k</sup> N* 

�*bk*| *RQ* = exp

Matching from the right with �*a<sup>N</sup>* | we obtain

We adopt the positive root

and equ. (32) becomes

Inversely we have the decomposition

The relation *Q* receives the decomposition

Consider now

*<sup>a</sup><sup>N</sup>* � <sup>=</sup> <sup>1</sup>

�*a<sup>N</sup>* | *b<sup>j</sup>*

 *bj* 

> *N*

> *k*=1

 *N*

*n*=1

*N N*

 *b<sup>j</sup>* 

*<sup>a</sup><sup>N</sup>* � <sup>=</sup> <sup>1</sup>

exp <sup>−</sup>2*πi jk N* 

exp <sup>2</sup>*πi mn N* 

�*bk*| *Q* = exp

<sup>2</sup>*πi*(*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) *N*

> <sup>2</sup>*πi <sup>k</sup> N*

*k*=1

*<sup>a</sup><sup>N</sup>* � <sup>=</sup> <sup>1</sup>

 1 *bj* *k*


*<sup>N</sup>* (33)



(39)

�*bk*+1| (40)

�*bk*+1| (41)

�*bk*| *Q* = �*bk*+1| (37) �*b<sup>N</sup>* | *Q* = �*b*1| (38)

<sup>√</sup>*<sup>N</sup>* (34)

$$Q^n R^m = \exp\left[2\pi i \frac{nm}{N}\right] R^m Q^n \tag{43}$$

In our discrete model the non-commutativity is determined by *N*. As *N* → ∞ the relation-operators *Q* and *R* commute. However it would be hasty to conclude that as *N* → ∞ we reach the continuum. The transition from the discrete to the continuum is a subtle affair and many options are available. Let us define

$$L = Na \qquad p = \frac{2\pi}{L} \tag{44}$$

Then

$$\exp\left[2\pi i \frac{1}{N}\right] = \exp\left[ipa\right].\tag{45}$$

What counts is the size of the available phase space and we may use Planck's constant *h* as a unit measuring the number of phase space cells. Using rather exp *<sup>i</sup>* ¯*<sup>h</sup> pa* , equ.(42) becomes

$$QR = \exp\left[\frac{i}{\hbar}pa\right]RQ\tag{46}$$

Approaching the continuum we may replace the discrete operators by exponential forms

$$R = \exp\left[\frac{i}{\hbar}pX\right] \tag{47}$$

$$Q = \exp\left[\frac{i}{\hbar}aP\right].\tag{48}$$

With *R* and *Q* unitary operators, *X* and *P* are hermitian operators. From equs. (46), (47), (48), we deduce

$$[X,P] = i\hbar.\tag{49}$$

The foundational non-commutative law of Quantum Mechanics testifies that there is a limit size ¯*h* ∼ *pa* in dividing the phase space. With *p* ∼ *mv* ≃ *mc* we understand that *a* represents the Compton wavelength.

### **4. Conclusions**

We are used first to wonder about particles or states and then about their interactions. First to ask about "what is it" and afterwards "how is it". On the other hand, quantum mechanics displays a highly relational nature. We are led to reorient our thinking and consider that things have no meaning in themselves, and that only the correlations between them are "real" [12]. We adopted the Peircean relational logic as a consistent framework to prime correlations and gain new insights into these theories. The logic of relations leads us naturally to the fundamental quantum rule, the probability as the square of an amplitude. The study of a simple discrete model, once extended to the continuum, reveals that only finite degrees of freedom can live in a given phase space. The "granularity" of phase space (how many cells reside within a given phase space) is determined by Planck's constant *h*.

[3] C. S. Peirce, *On the algebra of logic*, American Journal of Mathematics 3, pp 15-57

Relational Quantum Mechanics http://dx.doi.org/10.5772/54892 369

[4] S. Eilenberg and S. Maclane, *General theory of natural equivalences*, Transactions of

[5] F. Lawvere and S. Schanuel, *Conceptual Mathematics: A first indroduction to categories*,

[6] C. Isham, *A New Approach to Quantising Space-Time: I. Quantising on a General*

[7] A. Doring and C. Isham, *A topos foundation for Theoretical Physics: I. Formal*

[8] J. Baez and J. Dolan, *Higher-dimensional algebra and topological quantum field theory*,

[9] J. Baez, Quantum quandaries: A category-theoretic perspective, quant-ph/0404040 preprint, to appear in Structural Foundations of Quantum Gravity, Oxford University

[10] S. Abramsky and B. Coecke, *A categorical sematics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer

[12] Vlatko Vedral, *Quantum physics: Entanglement hits the big time*, Nature 425, 28-29 (4

[13] A. Nicolaidis and V. Kiosses, *Spinor Geometry,* Int. J. of Mod. Phys. A27, Issue 22, id.

[14] E. Cartan, *Les groupes projectifs qui ne laissent invariante aucune multiplicite*´ *plane*, Bull. Soc. Math. France 41, 53-96 (1913); E. Cartan, Leçons sur la th*e*´orie des spineurs,

[15] R. Penrose, *A spinor approach to General Relativity*, Annals of Physics, 10, 171 - 201 (1960); R. Penrose and W. Rindler, *Spinors and space - time*, Vol. 1, Cambridge

[16] A. Nicolaidis and V. Kiosses, Quantum Entanglement on Cosmological Scale, to appear

vols 1 and 2, Expos*e*´s de Geom*e*´trie, Hermann, Paris (1938)

the American Mathematical Society, 58, pp 239-294 (1945)

*Category*, Adv. Theor. Math. Phys. 7, 331-367 (2003)

*languages for physics*, J. Math. Phys. 49: 053515, 2008

Cambridge University Press (1997)

J. Math. Phys. 36, 6073 (1995)

[11] J. Schwinger, Proc. N.A. S. 45, 1542 (1959)

Science Press (2004)

September 2003)

1250126, arXiv:1201.6231

University Press (1984)

(1880)

Press

Discerning the foundations of a theory is not simply a curiosity. It is a quest for the internal architecture of the theory, offering a better comprehension of the entire theoretical construction and favoring the study of more complex issues. We have indicated elsewhere [13] that a relation may be represented by a spinor. The Cartan – Penrose argument [14, 15], connecting spinor to geometry, allowed us to study geometries using spinors. Furthermore we have shown that space-time may emerge as the outcome of quantum entanglement [16].

It isn't inappropriate to connect category theory and relational logic, the conceptual foundations of quantum mechanics, to broader philosophical interrogations. Relational and categorical principles have been presented by Aristotle, Leibniz, Kant, Peirce, among others. Relational ontology is one of the cornerstones of Christian theology, advocated consistently by the Fathers (notably by Saint Gregory Palamas). We should view then science as a "laboratory philosophy" and always link the meaning of concepts to their operational or practical consequences.

### **Acknowledgment**

This work has been supported by the Templeton Foundation.

### **Author details**

A. Nicolaidis

*<sup>⋆</sup>* Address all correspondence to: nicolaid@auth.gr

Theoretical Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

### **References**


8 Quantum Mechanics

**4. Conclusions**

practical consequences.

**Acknowledgment**

**Author details**

A. Nicolaidis

**References**

We are used first to wonder about particles or states and then about their interactions. First to ask about "what is it" and afterwards "how is it". On the other hand, quantum mechanics displays a highly relational nature. We are led to reorient our thinking and consider that things have no meaning in themselves, and that only the correlations between them are "real" [12]. We adopted the Peircean relational logic as a consistent framework to prime correlations and gain new insights into these theories. The logic of relations leads us naturally to the fundamental quantum rule, the probability as the square of an amplitude. The study of a simple discrete model, once extended to the continuum, reveals that only finite degrees of freedom can live in a given phase space. The "granularity" of phase space (how many cells

Discerning the foundations of a theory is not simply a curiosity. It is a quest for the internal architecture of the theory, offering a better comprehension of the entire theoretical construction and favoring the study of more complex issues. We have indicated elsewhere [13] that a relation may be represented by a spinor. The Cartan – Penrose argument [14, 15], connecting spinor to geometry, allowed us to study geometries using spinors. Furthermore we have shown that space-time may emerge as the outcome of quantum entanglement [16]. It isn't inappropriate to connect category theory and relational logic, the conceptual foundations of quantum mechanics, to broader philosophical interrogations. Relational and categorical principles have been presented by Aristotle, Leibniz, Kant, Peirce, among others. Relational ontology is one of the cornerstones of Christian theology, advocated consistently by the Fathers (notably by Saint Gregory Palamas). We should view then science as a "laboratory philosophy" and always link the meaning of concepts to their operational or

Theoretical Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

[1] A. Nicolaidis, *Categorical Foundation of Quantum Mechanics and String Theory,* Int.

[2] C. S. Peirce, *Description of a notation for the logic of relatives, resulting from an amplification of the conceptions of Boole's calculus of logic*, Memoirs of the American

reside within a given phase space) is determined by Planck's constant *h*.

This work has been supported by the Templeton Foundation.

*<sup>⋆</sup>* Address all correspondence to: nicolaid@auth.gr

J. Mod. Phys. A24: 1175 - 1183, 2009

Academy of Sciences 9, pp 317-378 (1870)


**Chapter 17**

**On the Dual Concepts of**

Cynthia Kolb Whitney

http://dx.doi.org/10.5772/53326

**1. Introduction**

**'Quantum State' and 'Quantum Process'**

Many of us who made a living in the 20th century did so by functioning as some kind of engineer. Though schooled mostly in Physics, this author often functioned in those days as an engineer. It was a good continuing education. One aspect of it was the big tool kit in use. For example, some subject systems were best viewed in the frequency domain: a system function‐ ing as a filter would suppress some frequencies and enhance others. But other systems were better viewed in the time domain: a system functioning as a controller would take a time series of input signals and produce a time series of output commands. Neither approach was considered more right, or more fundamental, than the other. They were complementary.

But in the 20th century, things felt less eclectic in Physics. Especially in the literature of Quantum Mechanics (QM), there often seemed to be a lot of passion about what viewpoint was allowed, and what viewpoint was not allowed. We were taught that it just was not correct to think of an atom as a nucleus with electrons in orbits around it. There could *not* be orbits; there *had to* be only 'orbitals', a new word coined to refer to complex wave functions that extended over all space, and provided only spatial densities of probability, in the form of squared amplitude. Except for its phase factor, there was no sense of time-line to an orbital. It was a stable state.

So in QM, the emphasis was all on the stable states. Between the stable states, there could occur transitions, resulting in emission or absorption of a photon, but the state transitions themselves were essentially instantaneous, and not open to study. This emphasis on the stable states, and the avoidance of the transitions between them, implied that questions about the details of state

Back at the turn of the 20th century, there was a good reason for the avoidance of details about process in QM: we did not understand how any atom could resist one totally destructive

> © 2013 Whitney; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Whitney; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

transitions should be regarded as illegitimate.

### **Chapter 17**
