**9. Math and temporal (t > 0) space duality**

Every physical science existed within our temporal subspace must be temporal (i.e., t > 0); otherwise, it is a virtual (or fictitious) science as mathematics does. The burden of a scientific postulation is to prove it exists within our universe and then find the solution. We shall now show that there exists a duality between science and mathematics in which any scientific hypothesis has to be shown that it is within the boundary condition of our temporal universe, before accepting it as a real postulation. Otherwise, the hypothesis is not a guarantee to be physically real. One of the essential boundary conditions is the causality condition (i.e., t > 0), which is to show that the solution is temporal and causal (i.e., t > 0). For instance, take Einstein's energy equation [11] as an example as given by.

$$\mathfrak{e} = \mathfrak{m}\mathfrak{c}^2\tag{2}$$

where m is the rest mass and c is the speed of light. In view of this equation, we first see that it is not a temporal or time-domain function. Strictly speaking, this equation cannot be directly implemented within our temporal subspace, since our universe is a temporal variable spatial function which can be described by [1, 2].

$$\mathbf{f}(\mathbf{x}, \mathbf{y}, \mathbf{z}; \mathbf{t}), \mathbf{t} > \mathbf{0} \tag{3}$$

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**Figure 7.**

*space (d).*

*Schrödinger's Cat and His Timeless (t = 0) Quantum World*

space. Since energy equation of Eq. () is not time variable equation, it is apparent that the equation cannot be directly implemented within our temporal universe. To make the energy equation be acceptable or match to our temporal (i.e., t > 0) subspace condition, we can transform the equation to become time-domain or

<sup>∂</sup>*<sup>t</sup>* <sup>=</sup> *<sup>c</sup>*<sup>2</sup> \_\_\_\_\_ <sup>∂</sup>*m*(*t*)

where ∂ε(*t*)/∂*t* is the rate of increasing energy conversion, ∂*m*(*t*)/∂*t* is the corresponding rate of mass reduction, *c* is the speed of light, and t > 0 represents a forward time variable. Notice that we have transformed the equation into a partial differential form which exists only at time t > 0. This indicates that the solution as obtained by this equation is compiled by means of the causality (i.e., t > 0) constraint, of which the solution can be used within our temporal universe (i.e., t > 0). On the other hand, if Eq. (4) is imposed by a timeless (i.e., t = 0) constraint as

<sup>∂</sup>*<sup>t</sup>* <sup>=</sup> *<sup>c</sup>*<sup>2</sup> \_\_\_\_\_ <sup>∂</sup>*m*(*t*)

then we see that the solution as obtained by Eq. (5) will be timeless (i.e., existed at t = 0). And it cannot be implemented within our temporal (i.e., t > 0) universe. Needless to say, if we put a constraint on Eq. (3) as can be shown by f(x, y, z; t), t = 0. Then we see that a temporal equation has been transformed into a timeless equation which exists only at t = 0, in which we see that Eq. (5) cannot be used

As we know that a timeless space is actually a mathematical virtual space, only mathematician and possibly quantum physicist can produce it, since quantum mechanics is mathematics. Nevertheless, a timeless space has no time and no substance in it. When we look back at all the fundamental laws in science, they are mostly presented by point-singularity approximation, and many of them are timeless or time-independent equations, such as Schrödinger's equation. And we have shown in proceeding that Schrodinger's quantum machine is timeless since its mechanics was built on an empty subspace. Nevertheless we are going to show some possible outcome when a timeless superposition principle is implemented within a timeless platform. Before showing, let us introduce a few subspaces that may be

*This figure shows an absolute empty space (a), a virtual space (b), a Newtonian space (c), and Yu's temporal* 

<sup>∂</sup>*<sup>t</sup>* ,t <sup>&</sup>gt; <sup>0</sup> (4)

<sup>∂</sup>*<sup>t</sup>* ,t <sup>=</sup> <sup>0</sup> (5)

*DOI: http://dx.doi.org/10.5772/intechopen.86970*

temporal equation as given by [].

shown by

∂ε(*t*) \_\_\_\_\_

∂ε(*t*) \_\_\_\_\_

within our temporal universe (i.e., t > 0).

used for the illustration, as depicted in **Figure 7**.

where (x, y, z) is a spatial variable and (t > 0) is a forward time variable, in which we see that every subspace within our universe is time-dependent variable *Advances in Quantum Communication and Information*

tion such as timeless (i.e., t = 0) superposition.

**9. Math and temporal (t > 0) space duality**

Einstein's energy equation [11] as an example as given by.

states computing [10], then the superposition principle as derived from the timeindependent Schrödinger equation would have a problem, as applied within our temporal universe, since the superposition is timeless. For example, those instant and simultaneous response promises by the fundamental principle do not exist within our temporal space. And the postulated Schrödinger's cat is not a physical realizable solution, in which we have shown that the burden of the cat's half-life can be liberated by using a temporal (i.e., t > 0) radioactive particle instead, in which we see that the paradox of Schrödinger's cat may never be discovered that it is not a paradox, if we did not discover that Schrödinger's quantum mechanics is timeless. Since the Schrödinger equation is a timeless quantum computer, which is designed to solve a variety of particle's quantum dynamics, the solution as obtained from Schrodinger's equation is also timeless, which produces a non-realizable solu-

We see that if one forces a timeless (i.e., t = 0) solution into a temporal (i.e., t > 0) subspace, one would anticipate paradox solution that does not exist within our temporal universe, such as Schrödinger's half-live cat. This is equivalent to chasing a ghost of a timeless half-life cat in a temporal subspace, in which we have found that a timeless radioactive particle was inserted within Schrödinger's box! As to answer the "yes" part, if temporal aspect as applying a quantum mechanical solution is not an issue within our temporal space, then we have already seen scores of solutions as obtained from the Schrödinger equation which have been brought to use in practice, since the birth of quantum mechanics in 1933. This is similar to using mathematics (i.e., a timeless machine) to obtain solution for timedependent application and sometime produces solution not physically realizable, in which we see that the Schrödinger equation is a mathematics, which requires a time boundary condition (i.e., t > 0) to justify that its solution is physically realizable.

Every physical science existed within our temporal subspace must be temporal (i.e., t > 0); otherwise, it is a virtual (or fictitious) science as mathematics does. The burden of a scientific postulation is to prove it exists within our universe and then find the solution. We shall now show that there exists a duality between science and mathematics in which any scientific hypothesis has to be shown that it is within the boundary condition of our temporal universe, before accepting it as a real postulation. Otherwise, the hypothesis is not a guarantee to be physically real. One of the essential boundary conditions is the causality condition (i.e., t > 0), which is to show that the solution is temporal and causal (i.e., t > 0). For instance, take

= mc<sup>2</sup> (2)

where m is the rest mass and c is the speed of light. In view of this equation, we first see that it is not a temporal or time-domain function. Strictly speaking, this equation cannot be directly implemented within our temporal subspace, since our universe is a temporal variable spatial function which can be described by [1, 2].

f(x,y, z;t),t > 0 (3)

where (x, y, z) is a spatial variable and (t > 0) is a forward time variable, in which we see that every subspace within our universe is time-dependent variable

**116**

space. Since energy equation of Eq. () is not time variable equation, it is apparent that the equation cannot be directly implemented within our temporal universe. To make the energy equation be acceptable or match to our temporal (i.e., t > 0) subspace condition, we can transform the equation to become time-domain or temporal equation as given by [].

$$\frac{\partial \mathbf{c}(t)}{\partial t} = \mathcal{L}^2 \frac{\partial m(t)}{\partial t}, \mathbf{t} > \mathbf{0} \tag{4}$$

where ∂ε(*t*)/∂*t* is the rate of increasing energy conversion, ∂*m*(*t*)/∂*t* is the corresponding rate of mass reduction, *c* is the speed of light, and t > 0 represents a forward time variable. Notice that we have transformed the equation into a partial differential form which exists only at time t > 0. This indicates that the solution as obtained by this equation is compiled by means of the causality (i.e., t > 0) constraint, of which the solution can be used within our temporal universe (i.e., t > 0).

On the other hand, if Eq. (4) is imposed by a timeless (i.e., t = 0) constraint as shown by

$$\frac{\partial \mathbf{c}(t)}{\partial t} = \mathcal{L}^2 \frac{\partial m(t)}{\partial t}, \mathbf{t} = \mathbf{O} \tag{5}$$

then we see that the solution as obtained by Eq. (5) will be timeless (i.e., existed at t = 0). And it cannot be implemented within our temporal (i.e., t > 0) universe.

Needless to say, if we put a constraint on Eq. (3) as can be shown by f(x, y, z; t), t = 0. Then we see that a temporal equation has been transformed into a timeless equation which exists only at t = 0, in which we see that Eq. (5) cannot be used within our temporal universe (i.e., t > 0).

As we know that a timeless space is actually a mathematical virtual space, only mathematician and possibly quantum physicist can produce it, since quantum mechanics is mathematics. Nevertheless, a timeless space has no time and no substance in it. When we look back at all the fundamental laws in science, they are mostly presented by point-singularity approximation, and many of them are timeless or time-independent equations, such as Schrödinger's equation. And we have shown in proceeding that Schrodinger's quantum machine is timeless since its mechanics was built on an empty subspace. Nevertheless we are going to show some possible outcome when a timeless superposition principle is implemented within a timeless platform. Before showing, let us introduce a few subspaces that may be used for the illustration, as depicted in **Figure 7**.

#### **Figure 7.**

*This figure shows an absolute empty space (a), a virtual space (b), a Newtonian space (c), and Yu's temporal space (d).*

In Ref. to this figure, we see an absolute empty space which has no time, no substance, and no coordinate. A mathematical virtual space is an empty and timeless space with spatial coordinates. A Newtonian space is filled with substance but treated time as an independent variable. And finally a temporal space is filled with substance and existed only at t > 0, of which substance and time coexisted [1, 2]. We further see that none of the spaces such as absolutely empty, virtual, and Newtonian spaces can be a subspace of the temporal space or vice versa, since temporal (i.e., t > 0) space is a time-invariant system (i.e., the system analysis standpoint) and the others are not.

Now, let us take an example as illustrated in **Figure 8** in which we assume three delta functions (t−t1), (t−t2), and (t−t3) representing a set of particles that are plunging into a timeless subspace system diagram as depicted in **Figure 8b**. We see that output delta functions are superimposed on top of each other at t = 0, shown in **Figure 8c**, of which we note that all the input pulses (i.e., particles) lost their temporal identities within a timeless space. And this is precisely the superposition principle tells us that the entire quantum states exist simultaneously and instantly (i.e., at t = 0). However, superposition principle does not exist within a temporal (i.e., t > 0) space. Since time is distance and distance is time, the entire quantum states exist simultaneously everywhere only within a timeless space as can be seen in **Figure 8e**. Therefore, it is a serious mistake to assume superposition principle works within our temporal universe, such as the paradox of Schrödinger's cat and possibly others. It is interesting to find out from system analysis standpoint [3] how a timeless (i.e., t = 0) subspace respond to a time-domain input excitation.

On the other hand, if we plunge the delta pulses within a temporal subspace, as shown in **Figure 9**, we see the output responses are faithfully temporally reproduced, which shows the time-invariant property of our temporal subspace, in which these particles (e.g., quantum states) are temporally separated, instead of superposing together at t = 0. And this is precisely the moment when we open Schrodinger's box, we found the cat can only be either dead or alive but not both at the same time. Instead of assuming the fundamental principle collapses to justify the superposition principle.

#### **Figure 8.**

*(a) Shows a set of three pulses (e.g., particles) within a temporal subspace as shown in topographical view in (d). As these particles plunge into a timeless subspace of (b), the output responses are superposing at t = 0 shown in (c), and the superimposed particles can be found all over the timeless domain as can be seen in (e). It is interesting to note that within a timeless space, all things are in one and one is everywhere within the space.*

**119**

*space.*

**Figure 10.**

**Figure 9.**

*The time-invariant response property from a temporal subspace.*

*Schrödinger's Cat and His Timeless (t = 0) Quantum World*

In summing up our illustration, our universe is a causal (i.e., t > 0) timeinvariant system which can be symbolically described by f(x, y, z; t), t > 0, in which time and space coexisted. Since time is a constant forwarded variable, the speed of time is determined by the velocity of light as given by t ≈ 1/c, where c ≈ 186,282 miles/sec, by which our temporal universe was indeed created by means of Einstein energy equation that was derived with his relativity theory, in which we see that time is distance and distance is time within our temporal universe. In contrast within a timeless (i.e., t = 0) space, it has no time and no distance, since d = ct and t = 0, for which everything collapses instantly at t = 0 (or d = 0) within a timeless space, as superposition principle does. Although scores of quantum mechanical solutions have been put into use, it is the fundamental principle of superposition that confronted with the temporal boundary condition

Regardless the mutual exclusive issues between timeless and temporal subspaces, some quantum scientists still believe they can implant superposition principle within our universe. This is the reason that we would show what would happen when a multi-quantum states particle is implemented within a temporal space. For simplicity, we will simulate a two-quantum states particle plunging into an empty subspace as shown in **Figure 10a** and **b**. We further let two quantum states associated with two eigenfunctions exp.[i(ω1t)] and exp.[i(ω2t)], where ω represents the angular frequency of the quantum state. And the output response from an empty space is given in **Figure 10c** that corresponds to a "timeless" superposing dualquantum state (a real quantity), where we assume energy is conserved. When this timeless simulated response is plunged into a temporal (i.e., t > 0) space as depicted in **Figure 10d**, its output response is shown in **Figure 10e**, in which we note that the output response occurs at t > 0 and it was not started instantly at t = 0, since time is distance and distance is time within a temporal space. In view of this simulation, we learn that particle's quantum states lost their personalities as soon it plunges into a timeless space. Since the timeless subspace is assumed to be within a temporal (i.e., t > 0) space, it is the temporal space that dictates the end response, as shown in **Figure 10e**. This shows us that all the "instance and simultaneous" quantum states as indicated by the superposition principle are not happening. Equivalently

*System simulation for an empty subspace within a temporal space. (a) Input excitation, (b) empty timeless system, (c) output response from an empty space, (d) temporal system, and (e) final output from a temporal* 

*DOI: http://dx.doi.org/10.5772/intechopen.86970*

t > 0 that produces Schrödinger's cat.

*Advances in Quantum Communication and Information*

standpoint) and the others are not.

In Ref. to this figure, we see an absolute empty space which has no time, no substance, and no coordinate. A mathematical virtual space is an empty and timeless space with spatial coordinates. A Newtonian space is filled with substance but treated time as an independent variable. And finally a temporal space is filled with substance and existed only at t > 0, of which substance and time coexisted [1, 2]. We further see that none of the spaces such as absolutely empty, virtual, and Newtonian spaces can be a subspace of the temporal space or vice versa, since temporal (i.e., t > 0) space is a time-invariant system (i.e., the system analysis

Now, let us take an example as illustrated in **Figure 8** in which we assume three delta functions (t−t1), (t−t2), and (t−t3) representing a set of particles that are plunging into a timeless subspace system diagram as depicted in **Figure 8b**. We see that output delta functions are superimposed on top of each other at t = 0, shown in **Figure 8c**, of which we note that all the input pulses (i.e., particles) lost their temporal identities within a timeless space. And this is precisely the superposition principle tells us that the entire quantum states exist simultaneously and instantly (i.e., at t = 0). However, superposition principle does not exist within a temporal (i.e., t > 0) space. Since time is distance and distance is time, the entire quantum states exist simultaneously everywhere only within a timeless space as can be seen in **Figure 8e**. Therefore, it is a serious mistake to assume superposition principle works within our temporal universe, such as the paradox of Schrödinger's cat and possibly others. It is interesting to find out from system analysis standpoint [3] how

a timeless (i.e., t = 0) subspace respond to a time-domain input excitation.

On the other hand, if we plunge the delta pulses within a temporal subspace, as shown in **Figure 9**, we see the output responses are faithfully temporally reproduced, which shows the time-invariant property of our temporal subspace, in which these particles (e.g., quantum states) are temporally separated, instead of superposing together at t = 0. And this is precisely the moment when we open Schrodinger's box, we found the cat can only be either dead or alive but not both at the same time. Instead of assuming the fundamental principle collapses to justify the superposition

*(a) Shows a set of three pulses (e.g., particles) within a temporal subspace as shown in topographical view in (d). As these particles plunge into a timeless subspace of (b), the output responses are superposing at t = 0 shown in (c), and the superimposed particles can be found all over the timeless domain as can be seen in (e). It is interesting to note that within a timeless space, all things are in one and one is everywhere within the space.*

**118**

**Figure 8.**

principle.

In summing up our illustration, our universe is a causal (i.e., t > 0) timeinvariant system which can be symbolically described by f(x, y, z; t), t > 0, in which time and space coexisted. Since time is a constant forwarded variable, the speed of time is determined by the velocity of light as given by t ≈ 1/c, where c ≈ 186,282 miles/sec, by which our temporal universe was indeed created by means of Einstein energy equation that was derived with his relativity theory, in which we see that time is distance and distance is time within our temporal universe. In contrast within a timeless (i.e., t = 0) space, it has no time and no distance, since d = ct and t = 0, for which everything collapses instantly at t = 0 (or d = 0) within a timeless space, as superposition principle does. Although scores of quantum mechanical solutions have been put into use, it is the fundamental principle of superposition that confronted with the temporal boundary condition t > 0 that produces Schrödinger's cat.

Regardless the mutual exclusive issues between timeless and temporal subspaces, some quantum scientists still believe they can implant superposition principle within our universe. This is the reason that we would show what would happen when a multi-quantum states particle is implemented within a temporal space. For simplicity, we will simulate a two-quantum states particle plunging into an empty subspace as shown in **Figure 10a** and **b**. We further let two quantum states associated with two eigenfunctions exp.[i(ω1t)] and exp.[i(ω2t)], where ω represents the angular frequency of the quantum state. And the output response from an empty space is given in **Figure 10c** that corresponds to a "timeless" superposing dualquantum state (a real quantity), where we assume energy is conserved. When this timeless simulated response is plunged into a temporal (i.e., t > 0) space as depicted in **Figure 10d**, its output response is shown in **Figure 10e**, in which we note that the output response occurs at t > 0 and it was not started instantly at t = 0, since time is distance and distance is time within a temporal space. In view of this simulation, we learn that particle's quantum states lost their personalities as soon it plunges into a timeless space. Since the timeless subspace is assumed to be within a temporal (i.e., t > 0) space, it is the temporal space that dictates the end response, as shown in **Figure 10e**. This shows us that all the "instance and simultaneous" quantum states as indicated by the superposition principle are not happening. Equivalently

**Figure 9.**

*The time-invariant response property from a temporal subspace.*

#### **Figure 10.**

*System simulation for an empty subspace within a temporal space. (a) Input excitation, (b) empty timeless system, (c) output response from an empty space, (d) temporal system, and (e) final output from a temporal space.*

speaking, this is precisely why the dual-quantum states of the radioactive particle within Schrodinger's box are dysfunctional or impaired, within a temporal space.
