Model Validation

The validation was based on the model calibration process through the comparison of monitored and simulated data from the living room. Nowadays, this is frequently used in research as a means of validating results by calibrating the model through the comparison of monitored data in short periods of time and building simulations to identify thermal comfort and energy benefits [8,24–26]. Monitored data at various points in the building were used for model calibration (Figure 6). The indoor air temperature of the main room was used as the parameter for model validation, given its strong connection to the thermal environment of the entire building. The simulated temperature in the main room was compared to the values recorded during observation periods (Figure 18). The comparison evaluated indicators used by the scientific community and defined by international organizations such as the American Society of Heating, Refrigerating and Air Conditioning (ASHRAE) and the National Renewable Energy Laboratory (NREL) [27], which establish acceptable ranges for statistical calibration rates. After calibration, hourly and monthly

simulation data for an entire year could be obtained. A model is considered valid if the difference between the simulated and monitored values falls within the acceptable range defined by international standards (RMSE 10% and NBME 25%) [28,29]. Two well-accepted statistical indicators for model validation are the normalized mean bias error (NBME) and the root mean square error of the bias (RMSE). To assess the correlation between the simulation and monitoring results, specific criteria were adopted. These criteria are formally defined as follows:

$$NMBE\left(\frac{0}{0}\right) = \frac{\sum\_{i=1}^{n} \left(t\_{ip} - t\_{mi}\right)}{n - 1} \times \frac{1}{\frac{t\_{m}}{\underline{t\_{m}}}} \times 100$$

$$CV(RMSE)\left(\frac{0}{0}\right) = \sqrt{\frac{\sum\_{i=1}^{n} \left(t\_{ip} - t\_{mi}\right)^{2}}{n - 1}} \times \frac{1}{\frac{t\_{m}}{\underline{t\_{m}}}} \times 100$$

**Figure 18.** Temperature difference between monitored and simulated values in the main room.

In this study, the simulated temperature values (*tip*) of node *i* were compared to the monitored temperature values (*tmi*) of node *i*, where the arithmetic means of a sample of *n* measured temperature data (*tm*) were calculated, and *n* was the number of monitored temperature data during the monitoring period. The normalized mean bias error (NMBE) indicator provided information on the difference between the simulated and measured temperatures. A positive NMBE value indicated that the simulated temperatures were higher than the monitored data, whereas a negative value indicated that the simulated temperatures were lower than the monitored temperatures. The ideal NMBE value was zero.

The coefficient of variation of the root mean square error (CV(RMSE)) indicator quantifies the relative correlation, expressed as a percentage, between the differences (simulated temperatures minus monitored temperatures) and the average of all the monitored temperatures [24,26]. The CV(RMSE) value is always positive, and an ideal value would be zero, indicating optimal accuracy. NMBE values, on the other hand, indicate the presence of systematic error or bias in the simulation. In contrast, CV(RMSE) serves as an indicator of precision, reflecting how closely the simulated results align with the monitored data. The temperature differences between the monitored and simulated temperatures in the main room are presented in Table 5. Additionally, the coefficient of variation of the root mean square error (CV(RMSE)) was calculated, which was 4.46%, and the normalized mean bias

error (NMBE) was found to be 4.28%. Both values were below the maximum validation threshold of 25% recommended by the ASHRAE [28].

**Table 5.** Temperature differences between monitored and simulated data in the main room. NMBE and CV (RMSE) values.

#### *3.3. Assessment of Annual Thermal Performance*

A comprehensive one-year simulation of the building was conducted using the validated model and its corresponding input parameters for annual simulations. In order to examine the dynamic free-running thermal behavior of the living room environment, the impact of free gains was deliberately disregarded.

Figure 19 presents the annual differences in the temperature of the simulated main living room and the outside temperature. It can be observed that the outdoor temperatures are representative of cities located at altitudes between 2000 and 3000 m above sea level in the Andean Mountain region on the Equatorial Line (such as the city of Azogues), which have a cold climate that affects the indoor temperatures of buildings. Regarding the indoor thermal comfort range between 18 ◦C and 22 ◦C, it can be observed that the simulated temperatures are mostly below the range. The number of hours per year that the building is below 18 ◦C is 7.338 h, that below 16 ◦C is 6.207 h, and that below 10 ◦C is 538 h. These values show that the indoor environment of the building, despite having temperatures below the comfort range, never reaches extreme cold temperatures or below 0 ◦C, which is

a characteristic of buildings that do not have active heating systems. This has caused the residents to adapt to lower temperatures and compensate for them with clothing [20].

**Figure 19.** Variations in air temperature of the main room over one year.

Figure 20 presents data on the relative humidity throughout one year at an hourly resolution. The humidity ratio in the main living room ranged from 4 to 11.7 g of water vapor per kilogram of dry air. The upper limit of relative humidity was set at 70%, indicating that the relative humidity values did not exceed this threshold throughout the evaluation period.

**Figure 20.** Variations in relative humidity of the main room over one year.

In Figure 19, the dashed line of simulated temperatures for one year in the living room shows the range obtained, with indoor temperatures ranging from 15 ◦C to 18 ◦C and relative humidity between 2 and 9 g of water vapor per kg of dry air, with an acceptable upper limit of 19 g of water vapor per kg of dry air and maximum acceptable relative humidity of 70%. This indicates that the traditional courtyard houses in the city of Azogues adapt well to the local climatic conditions, primarily due to their thermal masses. Additionally, the higher simulated temperatures in the living room compared to the exterior, even at night, demonstrate that the case study has passive design strategies that improve the indoor temperatures. This temperature value could potentially improve if the U-value of the thermal transmittance of the building's envelope decreases to 2.35, as stipulated by local regulations. In this way, the internal thermal gains would be retained, and more appropriate thermal comfort standards could be achieved.

If we analyze the annual thermal performance of the building with the temperature and relative humidity data from the living room, we can verify that the simulated temperature in the main room exceeds the 16 ◦C recorded during the monitoring period. Additionally, the annual relative humidity of the simulated main room matches the relative humidity recorded in the same space at 70%, allowing the building's occupants to adapt to the indoor environment despite having slightly lower temperatures than those recognized as the comfort range.


4.1.2. BIM-Specific Requirements

Tenderers deliver a substantial design effort in preparing proposals, especially in BIM projects which necessitate extra efforts in the early design stage, and stipends compensate this effort. Stipends increase competition, bidding pool diversity, SME inclusion in public procurement and quality of proposals, while the design level of effort reduces cost growth, and their absence can lead contractors to not participate in the process. The appropriate level of a stipend is somewhat flexible, but the rule of thumb for a two-stage process is one-third of the design effort (Alleman et al. 2020).

Level 3 BIM, through integration of the team, may blur the levels of responsibility and enhance risk and liability (Azhar 2011). Risks must be identified and allocated, especially regarding responsibility for the accuracy and coordination of data as well as updating information in collaborative models (Porwal and Hewage 2013). The BIM addendum addresses the risk of project participants assuming contributions made by other parties are accurate and provides for claim waivers (Porwal and Hewage 2013). The BIM addendum also specifies that the participation of the contractor, subcontractors and suppliers in a model does not constitute design services, although this cannot be applied to DB because of its single point of responsibility, which means parties only assume their traditional roles (Currie 2014). CIC BIM Protocol limits project team members' liability by stating that there is no warranty to the integrity of electronic data transmission and no liability for corruption or alteration occurring after transmission (Mosey et al. 2016).

Traditional insurance is not easily adaptable to BIM level 3, and new insurance products better tailored to collaborative projects are needed (Currie 2014). One possible solution is Integrated Project Insurance (IPI), which can be used in alliancing or IPD models. This model covers all major parties under one single policy and includes all the insurances needed for infrastructure projects (Currie 2014).

## 4.1.3. Dysfunctions of Current Rules

Regulatory stipends do not reflect the actual work carried out by tenderers in BIM projects, and public bodies must use the derogation procedure to offer better compensation. Absence of stipends can disinterest firms and competition and favor firms with stronger financial records to the detriment of smaller ones. Risks are allocated and not shared, which does not reflect the collaborative BIM process and furthers the silo effect. Furthermore, Quebec's contractual documents do not provide for BIM-specific liability limitations or claim waivers.


3.4.2. Control Design

To control and ensure the compensation of the EVSE reactive power consumption, the EBox communicates with a grid analyzer to know the reactive power value and send the calculated set-point to the emulator AFE, which generates the desired current. As shown in Figure 9, there are three different grid analyzer configurations: in the output of the V2G charger under test, in the Point of Common Coupling (PCC) of the facility or in both (V2G and PCC).

Depending on the location of the analyzer, the control algorithm implemented in the EBox has to be changed, achieving different performances:

reactive power of the charger is measured and directly compensated at the output of the regulator. In order to avoid the integration of the error produced by the charger, the derivative of this measure is compensated in the input of the regulator. In this way, the controller achieves a better time response capability. However, it has to be highlighted that the use of two grid analyzers increases close to 2% the final price of the solution.

**Figure 9.** Representation of the two possible locations of the grid analyzer in order to compensate the reactive power injected by the EVE.

**Figure 10.** Three types of different reactive compensation control strategies: (**a**) Open loop control, measuring reactive power at the output of the V2G charger. (**b**) Closed loop control, measuring reactive power at the PCC of the facility. (**c**) Closed loop control with V2G compensation, measuring both V2G charger and the PCC of the facility.

Figure 11 shows the simulation in MATLAB/Simulink of the reactive power consumption of the three different types of control that have been shown in previous Figure 10. It can be seen that in Figure 11a without any external loads, the reactive consumption in the PCC with open control loop as well as with closed control loop with V2G compensation are the same, while with closed control loop the response is slower than the previous case. However, if there is reactive power consumption by an external load, Figure 11b shows different behaviour. On one hand, with open control loop the reactive power consumption at the PCC in steady-state is determined by the external loads. On the other hand, with closed control loop, the power at the PCC can be fully compensated by the EVE, with a better time response to the V2G power demand by compensating it using two grid analyzers.

Comparing the last two control modes in Figure 11b, it should be noticed that the reactive power error given by the abrupt change of the V2G power at time *t* = 32 s, is quickly balanced with the third type of reactive control strategy. The reason is that the compensation of the V2G power avoids the error integration in the PI control, improving the time response and performance of the reactive power control at the PCC.

**Figure 11.** Simulation of the reactive power consumption using the same V2G consumption profile with abrupt consumption changes. Two scenarios have been considered: (**a**) without power consumption of the external loads; (**b**) with power consumption of the external loads. In every scenario, all three different control strategies have been tested: open loop, closed loop, and closed loop with V2G compensation.


**4. Regulation of bacterial GDH gene expression**

296 Dehydrogenases

high temperature for life.

sites (NAD+, ADP, NAD(P)H) [19]

hypothesized than the latter acts specifically in arginine catabolism by converting glutamate, a product of the ammonia-producing arginine succinyl transferase (AST) pathway, into 2-OG, since it is allosterically modulated by arginine (positively) and citrate (negatively) [25]. Similarly, the only active GDH from *Bacillus subtilis* (RocG, NAD+ dependent) appears to be involved in arginine and proline catabolism [46]. On the other hand, despite the catabolic function assigned to NAD+-GDHs, the existence of an NAD+ specific GDH with an unusual biosynthetic role has been reported in the oral bacterium *Capnocytophaga ochraea* [35]. In this case, it was found that only the NAD+-GDH ammonium assimilating activity could be detected in cell free extracts, probably due to the high concentration of ammonium and ammonium precursors that can be found in the gingival crevicular fluid. Interestingly, GDHs have been shown to play a substantial and even predominant role in nitrogen assimilation in conditions of N2 fixation in the Gram positive bacteria *Bacillus macerans* and *Bacillus polymyxa* [44, 45]. Nitrogen fixation only occurs under extreme nitrogen-limiting conditions, when nitrogen from other sources is very scarce. In these conditions nitrogen is always assimilated using the GS/GOGAT pathway, since the Km of GS for ammonium is much lower than that of GDH. This is not the case in *B. macerans* and *B. polymyxa*, as in these organisms the GOGAT activity is much lower than GDH activity in nitrogen-fixing cells. A NAD+-GDH involved in glutamate fermentation has also been described in the anaerobic Gram-positive bacteria *Peptostreptococcus asaccharolyticus* [54]. In this organism, GDH is the first enzyme of the glutamate fermentation via the hydroxyglutarate pathway, and can represent as much as 10% of total protein when grown on glutamate. Very high levels of GDH production has also been reported in some hyperthermophilic archaea like *Pyrococcus furiosus* or some *Thermococcus* strains [5, 6, 71]. These preferentially biosynthetic enzymes represent an exceptionally high percentage of total soluble protein of the cell, in some cases up to 20%, which suggests an important role of these enzymes in these organisms at an extremely

Determination of the GDH enzymatic structure has allowed the elucidation of the mechanisms for allosteric regulation and negative cooperativity. The activity of glutamate dehydrogenase in animals is allosterically regulated by purine nucleoside phosphates and other metabolic intermediates. In brief, GTP and ATP are allosteric inhibitors whereas GDP and ADP are allosteric activators. Hence, a lowering of the energy charge accelerates the oxidation of amino acids. Their *in vivo* regulation may be dependent on the metabolic status of the cell, or in the tissue they are located. The intracellular compartmentalization of the cofactors, and the GDH itself, may also drive the reaction in one or the other direction. In vertebrate cells, GDH appears to be localized primarily in the mitochondrial matrix [1]. The effects of nucleosides on mammalian GDH are complex. For the bovine liver GDH, four binding sites per subunit have been described, being the active site (site I), the adenine nucleotide regulatory site (site II), the guanine nucleotide regulatory site (site III) and the reduced coenzyme regulatory site (site IV). Some substrate and effectors bind just the active site (glutamate, oxoglutarate, ammonia, NADP+) while others are able to bind two different The diverse roles of bacterial GDH in different organisms provide for a variety of regulatory mechanisms. Here we show a few examples for selected bacteria in which transcriptional regulation of GDH genes has been characterized.

### **4.1. Regulation of GDH synthesis in the enterobacteria**

Transcriptional regulation of the *gdhA* gene, encoding NADP-GDH, was first described in the diazotrophic enterobacterium *Klebsiella pneumoniae*, and then in *Escherichia coli* [73]. Transcription of *gdhA* is repressed in both enteric bacteria under nitrogen limitation by the general nitrogen control system (Figure 3,A). This is consistent with the fact that low affinity for ammonium limits the use of GDH for glutamate synthesis at ammonium concentrations below 1 mM. The enterobacterial general nitrogen control (Ntr) system is a very well characterized signal transduction and regulatory network encompassing seven elements: the alternative σ factor σ54, encoded by *rpoN*, the uridylyl transferase-uridylyl removing enzyme GlnD, two PII signal transduction proteins, GlnB and GlnK, the two-component system NtrB-NtrC and the LysR-type transcriptional regulator Nac. Nitrogen status is signalled by the intracellular pools of glutamine (indicative of nitrogen sufficiency), and 2-OG (indicative of nitrogen limitation). These signals are perceived by the PII proteins by means of their reversible GlnD-dependent uridylylation in response to decreased glutamine levels and by allosteric modulation via 2-oxoglutarate binding. Interaction with PII proteins in either their uridylylated or deuridylylated states is responsible for posttranslational regulation of the activities of a variety of proteins involved in nitrogen metabolism, including glutamine synthetase, nitrogenase and the sensor kinase/phosphatase NtrB. Deuridylylated GlnB promotes NtrB-dependent dephosphorylation of NtrC under nitrogen excess [2, 74]. In nitrogen-limiting conditions, phosphorylated NtrC (NtrC-P) activates multiple σ54 dependent promoters, controlling the expression of over one hundred genes related in *E. coli*  [75]. NtrC-regulated genes encode multiple functions related to nitrogen metabolism, including transport and utilization pathways for diverse nitrogen sources, the nitrogen fixation sensor-regulator pair NifL-NifA (in the diazotrophic *Klebsiella pneumoniae*), and the nitrogen regulator Nac. Transcription of Nac is initiated from an NtrC-activated σ54 dependent promoter and autorepressed [76, 77]. Nac in turn activates and represses a set of genes whose products are mostly related to nitrogen metabolism. Nac-activated genes include those involved in the catabolism of histidine, proline, urea and alanine, among others [78-80]. Notably, Nac represses the genes coding for the two enzymes that synthesize glutamate, *gdhA,* encoding GDH, and *gltAB*, encoding GOGAT [81]. Unlike most LysR-type transcriptional regulators, the activity of Nac is not modulated by the presence of a small ligand, and the protein is synthesized in its active form [79]. Thus, the nitrogen limitation dependency of Nac regulation exclusively reflects the increase in concentration under nitrogen limitation due to its transcriptional regulation. Nac is not present in many bacteria containing the Ntr system, and is conspicuously absent in the closely related and wellcharacterized *S. enterica* [82]. As the need for two different regulators (NtrC and Nac) has been questioned, Nac has been proposed to act as an "adaptor" that integrates genes transcribed from σ70-dependent promoters (which cannot be directly regulated by the activator of σ54-dependent promoters NtrC) into the general nitrogen control network [83].

Glutamate Dehydrogenases: Enzymology, Physiological Role and Biotechnological Relevance 299

"weak" repression is in fact the result of the antagonist action of Nac on ArgP-mediated activation [82, 85]. The role of lysine as a signal can be rationalized if one considers that glutamate must serve two roles in the cell, as both an amino group donor for over 80% of the nitrogenous compounds in the cell, and as a counterion to K+ in osmotic pressure homeostasis. In order to sense the amount of glutamate being used for biosynthesis, the pools of one or more amino group acceptor may be used instead of glutamate itself. Lysine is likely one of those nitrogenous compounds that serve as surrogates for glutamate in signaling the extent of glutamate overflow from osmotic pressure homeostasis into the biosynthetic pathways for other nitrogenous compounds [85]. Consistently, *gdhA*p activity is decreased in rich media containing amino acids [81], and lysine levels may be one of several signals that mediate the feedback regulation of *gdhA*p. However, the identity of

**Figure 3.** Regulatory circuits for bacterial glutamate dehydrogenase genes. The cartoons represent the known regulatory circuits for *K. pneumoniae* (A), *P. putida* (B), *P. aeruginosa* (C), *B. subtilis* (D), *S. coelicolor*  (E) and *C. glutamicum* (E). Sigma 54- or SigL-dependent promoters are displayed as white arrows. Other promoter types are displayed as black arrows. Question marks represent aspects that are not well

characterized. CCR: carbon catabolite repression.

other compounds that may fulfill this role is as of yet unknown.

Transcription of the *gdhA* gene is repressed by Nac under nitrogen limitation in both *E. coli*  and *K. pneumoniae* (in contrast, *gdhA* expression is not nitrogen-regulated in *S. enterica*). Nac exhibits two modes of transcriptional repression of the *gdhA*p promoter. "Weak" repression involves Nac binding as a dimer to the promoter region in a single site located at -100 to -75 relative to the transcriptional start site [84]. "Strong" repression is observed when a Nac tetramer is simultaneously bound to the aforementioned site, centered at -89, and a downstream site centered at +47. The proposed regulatory mechanism for "strong" repression involves Nac bending DNA, looping out the intervening region, and forming a repressor loop reminiscent of those formed by LacI and GalR at the *lac* and *gal* promoters, respectively [79].

In addition to Nac-mediated repression, *gdhA* is subjected to positive regulation by a second LysR-type transcriptional regulator, ArgP. ArgP is an activator whose activity is antagonized by lysine. ArgP interacts with the *gdhA* promoter region at a single site between -100 and -50. Lysine inhibits interaction of ArgP with this site, thus preventing activation [85]. Activation is also prevented by Nac interaction with the overlapping binding site centered at -89, as both regulatory proteins bind in a mutually exclusive fashion. Thus "weak" repression is in fact the result of the antagonist action of Nac on ArgP-mediated activation [82, 85]. The role of lysine as a signal can be rationalized if one considers that glutamate must serve two roles in the cell, as both an amino group donor for over 80% of the nitrogenous compounds in the cell, and as a counterion to K+ in osmotic pressure homeostasis. In order to sense the amount of glutamate being used for biosynthesis, the pools of one or more amino group acceptor may be used instead of glutamate itself. Lysine is likely one of those nitrogenous compounds that serve as surrogates for glutamate in signaling the extent of glutamate overflow from osmotic pressure homeostasis into the biosynthetic pathways for other nitrogenous compounds [85]. Consistently, *gdhA*p activity is decreased in rich media containing amino acids [81], and lysine levels may be one of several signals that mediate the feedback regulation of *gdhA*p. However, the identity of other compounds that may fulfill this role is as of yet unknown.

298 Dehydrogenases

respectively [79].

reversible GlnD-dependent uridylylation in response to decreased glutamine levels and by allosteric modulation via 2-oxoglutarate binding. Interaction with PII proteins in either their uridylylated or deuridylylated states is responsible for posttranslational regulation of the activities of a variety of proteins involved in nitrogen metabolism, including glutamine synthetase, nitrogenase and the sensor kinase/phosphatase NtrB. Deuridylylated GlnB promotes NtrB-dependent dephosphorylation of NtrC under nitrogen excess [2, 74]. In nitrogen-limiting conditions, phosphorylated NtrC (NtrC-P) activates multiple σ54 dependent promoters, controlling the expression of over one hundred genes related in *E. coli*  [75]. NtrC-regulated genes encode multiple functions related to nitrogen metabolism, including transport and utilization pathways for diverse nitrogen sources, the nitrogen fixation sensor-regulator pair NifL-NifA (in the diazotrophic *Klebsiella pneumoniae*), and the nitrogen regulator Nac. Transcription of Nac is initiated from an NtrC-activated σ54 dependent promoter and autorepressed [76, 77]. Nac in turn activates and represses a set of genes whose products are mostly related to nitrogen metabolism. Nac-activated genes include those involved in the catabolism of histidine, proline, urea and alanine, among others [78-80]. Notably, Nac represses the genes coding for the two enzymes that synthesize glutamate, *gdhA,* encoding GDH, and *gltAB*, encoding GOGAT [81]. Unlike most LysR-type transcriptional regulators, the activity of Nac is not modulated by the presence of a small ligand, and the protein is synthesized in its active form [79]. Thus, the nitrogen limitation dependency of Nac regulation exclusively reflects the increase in concentration under nitrogen limitation due to its transcriptional regulation. Nac is not present in many bacteria containing the Ntr system, and is conspicuously absent in the closely related and wellcharacterized *S. enterica* [82]. As the need for two different regulators (NtrC and Nac) has been questioned, Nac has been proposed to act as an "adaptor" that integrates genes transcribed from σ70-dependent promoters (which cannot be directly regulated by the activator of σ54-dependent promoters NtrC) into the general nitrogen control network [83].

Transcription of the *gdhA* gene is repressed by Nac under nitrogen limitation in both *E. coli*  and *K. pneumoniae* (in contrast, *gdhA* expression is not nitrogen-regulated in *S. enterica*). Nac exhibits two modes of transcriptional repression of the *gdhA*p promoter. "Weak" repression involves Nac binding as a dimer to the promoter region in a single site located at -100 to -75 relative to the transcriptional start site [84]. "Strong" repression is observed when a Nac tetramer is simultaneously bound to the aforementioned site, centered at -89, and a downstream site centered at +47. The proposed regulatory mechanism for "strong" repression involves Nac bending DNA, looping out the intervening region, and forming a repressor loop reminiscent of those formed by LacI and GalR at the *lac* and *gal* promoters,

In addition to Nac-mediated repression, *gdhA* is subjected to positive regulation by a second LysR-type transcriptional regulator, ArgP. ArgP is an activator whose activity is antagonized by lysine. ArgP interacts with the *gdhA* promoter region at a single site between -100 and -50. Lysine inhibits interaction of ArgP with this site, thus preventing activation [85]. Activation is also prevented by Nac interaction with the overlapping binding site centered at -89, as both regulatory proteins bind in a mutually exclusive fashion. Thus

**Figure 3.** Regulatory circuits for bacterial glutamate dehydrogenase genes. The cartoons represent the known regulatory circuits for *K. pneumoniae* (A), *P. putida* (B), *P. aeruginosa* (C), *B. subtilis* (D), *S. coelicolor*  (E) and *C. glutamicum* (E). Sigma 54- or SigL-dependent promoters are displayed as white arrows. Other promoter types are displayed as black arrows. Question marks represent aspects that are not well characterized. CCR: carbon catabolite repression.

### **4.2. Regulation of GDH synthesis in pseudomonads**

Little is known about the transcriptional regulation of GDH synthesis in most Gramnegative bacteria. However, the regulatory mechanisms behind *gdhA* regulation were recently revealed in the soil bacterium *Pseudomonas putida* [86]. Similarly to its enteric counterparts, *P. putida* represses *gdhA* expression during nitrogen-limited growth in a Ntr system-dependent fashion (Figure 3,B). *P. putida* harbors a simplified general nitrogen control system that lacks the major PII protein, GlnB, and the transcriptional regulator Nac. In this scheme, the only PII protein, GlnK, controls the phosphorylation/dephosphorylation balance of NtrC in response to nitrogen availability using mechanisms likely similar to those found in the enterics [87]. In the absence of Nac, NtrC has been shown to directly regulate some of the functions that are controlled by Nac in the enterics, including activation of *codB*  and *ureD*, encoding cytosine deaminase and urease respectively [88], and repression of the GDH gene *gdhA* [86]. NtrC binds the σ70-dependent *gdhA*p promoter region cooperatively at four different sites, centered at positions -118, -95, -21 and +12 relative to the transcriptional start. While simultaneous occupancy of all four sites yields the maximal levels of repression, only the promoter-proximal I and II sites were found to be absolutely required for repression. A mechanism based on a repressor loop involving four NtrC dimers has been proposed for negative regulation of the *gdhA*p promoter in *P. putida*. Although repression is partially sensitive to the phosphorylation state of NtrC *in vitro*, *in vivo* evidence indicated that phosphorylation of NtrC is not required for efficient repression of the *gdhAp* promoter. Since NtrC synthesis is itself nitrogen regulated *via* the *glnA*p promoter, the increase in repressor concentration under nitrogen limitation appears to be the major determinant of nitrogen-regulated expression of *gdhA* in *P. putida* [86].

Glutamate Dehydrogenases: Enzymology, Physiological Role and Biotechnological Relevance 301

The *Bacillus subtilis* genome contains two genes encoding GDHs, *rocG* and *gudB*. While RocG is an enzymatically active GDH, GudB is inactive, due to a duplication of three amino acid residues at its active center. Decryptification of *gudB* in a *rocG* background is achieved by high-frequency acquisition of a suppressor mutation consisting of the precise deletion of part of the 9-bp direct repeat that prevents activity [46, 47]. Both RocG and decryptified GudB are primarily catabolic dehydrogenases, and *de novo* glutamate synthesis in *B. subtilis*

Similarly to *P. aeruginosa* GdhB, RocG is not nitrogen-regulated and is linked to arginine and proline catabolism, as its expression is induced by arginine, ornithine and, to a lesser extent, proline. Transcription from the *rocG* promoter depends on the product of the *Bacillus subtilis sigL* gene, an ortholog of the alternative σ factor σ54 [91], and is activated by the UAS binding protein RocR in response to the presence of arginine, ornithine, or proline with the assistance of the arginine-dependent activator AhrC (Figure 3,D). SigL, RocR and AhrC also control transcription of the two operons, *rocABC* and *rocDEF*, involved in arginine conversion into glutamate. In that regard, RocG-dependent deamination of glutamate to 2 oxoglutarate can be viewed as the final step in the use of arginine, ornithine, and proline as carbon or nitrogen, providing rapidly metabolizable carbon- or nitrogen-containing compounds for biosynthesis [92]. Interestingly, inducibility of RocG synthesis by arginine

The most salient feature of the *rocG*p SigL-dependent promoter is the absence of an upstream activator sequence (UAS) for RocR. Instead, the UAS present at the *rocA*p promoter, located immediately downstream from the *rocG* coding sequence, is the *cis-*acting element essential for RocR-dependent activation of *rocG*p. This sequence has been shown to be active when placed upstream or downstream and as far as 15 kb away from the target promoter [93]. According to the general model of σ54-dependent promoter activation, RocR bound to its target sequence activates transcription by interacting with promoter-bound σ54- RNA polymerase by a mechanism that involves looping out of the intervening DNA sequences. The AhrC protein is also required for activation of *rocG* (as well as *rocABC* and *rocDEF*), and it apparently modulates the activity of RocR by means of protein-protein

Unlike the arginine utilization operons *rocABC* and *rocDEF*, *rocG* transcription is subjected to carbon catabolite repression in the presence of glucose. Such repression is mediated by the regulatory protein CcpA. A *cre* (catabolite responsive element) site is located at positions +39 and +51, and is required for CcpA binding and *rocG*p repression. CcpA binding at this location prevents transcription from the *rocG*p promoter, but also acts as a roadblock for low-level readthrough transcription from an upstream promoter, which is relevant in the absence of both glucose and the cognate inducers [92]. As in other CcpA-regulated genes, CcpA-mediated repression requires the assistance of the accessory proteins HPr and Crh.

A final feature of the *B. subtilis* GDH RocG worth mentioning is its role in the regulation of the assimilatory glutamate synthase operon, *gltAB*. RocG belongs to a group of proteins

**4.3. Regulation of GDH synthesis in** *Bacillus subtilis*

precludes growth on glutamate as the sole carbon source.

is performed exclusively by GOGAT.

contacts [94].

*Pseudomonas aeruginosa* encodes two GDH enzymes, a NAD-dependent GDH encoded by *gdhB*, primarily involved in glutamate deamination to 2-oxoglutarate, and a NADP-dependent GDH, encoded by *gdhA*, primarily involved in ammonium assimilation. The catabolic enzyme is linked to the AST pathway for aerobic utilization of arginine as a carbon source. Glutamate is the end product of the AST pathway, and NAD-GDH serves the purpose of connecting arginine catabolism to the central metabolism *via* 2-oxoglutarate. Transcription of *gdhB* is activated by the arginine regulatory protein ArgR in response to the presence of arginine, which is also an allosteric modulator of the enzyme activity (Figure 3,C). An ArgR binding site is found immediately upstream from the -35 box of the *gdhB* promoter. Induction of the expression of *gdhB* and activation of the encoded dehydrogenase by arginine serve to direct the flow of glutamate into the TCA cycle [25]. Conversely, transcription of *gdhA* is repressed by ArgR in the presence of arginine. Such repression could serve to minimize the operation of an energy-consuming futile cycle involving the simultaneous function of *gdhA* and *gdhB* when *P. aeruginosa* uses arginine as a carbon source. Repression of *gdhA* expression is exerted from a single ArgR binding site centered at position -41, and a simple steric hindrance mechanism has been proposed at this promoter [89]. Downregulation of *gdhA* expression by the Ntr system under nitrogen limitation has also been reported [90], but the factors and mechanisms involved are uncharacterized. Since *P. aeruginosa* also lacks Nac, direct repression by NtrC similar to that observed in *P. putida* [86], may also occur in this organism.


4. The extended Feynman path integral and quantum measurement

#### 4.1. Why is it concerning with the Feynman path integral?

As we know, in the history of the quantum theory, there are three equivalent expressions, namely, the differential equation of Schrödinger, the matrix algebra of Heisenberg and the path integral formulation of Feynman. However, these three expressions have their own focuses. The Schrodinger and Heisenberg expressions focus on the evolution of states and operations, respectively, whereas the path integral formulation of Feynman on the "correlation" of point to point as states is evolving [11]. On the other hand, in quantum mechanics, when do a measurement on a wave function diffusing in all of space, such as the measurement of the position of an electron in the experiment of double-slit interference, we will find that the whole wave function will instantaneously collapse to this position measured with some probabilities. Obviously there may be some inner "correlation" in wave function transferring the action of the measurement from local part to whole. These two "correlations" have common characters and may be unified to be one.

Moreover, we notice that the action integral in Feynman path integral formulation is the classical form. The classical physics is born to be a local theory and of course cannot exhibit the character of nonlocality. However, the relativity theory is different. In relativity theory, the time and space are coupling. Beyond the light cones in Minkowski space, the space-time causality is broken, and this may cause the nonlocality. The superluminal velocities are forbidden in real world, but for a connection description of virtual paths in the path integral theory, it might be practicable. What will happen when we extend the classical action to relativistic action? Could the superluminal trajectories included in possible paths to calculate quantum amplitude in the Feynman theory cause the nonlocality? How is the relationship between "unitary evolution operation" and "quantum measurement"? These questions will be revealed when we extend the Feynman path integral.

#### 4.2. How to extend the Feynman path integral?

The formulation for Feynman path integral can be written as

$$K(r, r\_0; t, t\_0) = \mathbb{C} \sum\_{\text{all paths}} \exp\left(i\mathbb{S}/\hbar\right) \tag{1}$$

where the coefficient C is a constant independent of paths and S is the action with classical form

$$S(t\_0, t\_1) = \int\_{t\_0}^{t\_1} L(\dot{r}(t), r(t))dt\tag{2}$$

K rð Þ ;r0; t; t0 in Eq. (1) is the propagator and defined into

$$K(r, r\_0; t, t\_0) = \left\langle \left\langle r \middle| \hat{\mathcal{U}}(t, t\_0) \middle| r\_0 \right\rangle \right\rangle \tag{3}$$

Eq. (1) reveals an important assumption in Feynman path integral: the weights of different paths for propagator are the same. This assumption makes Feynman path integral very successful in nonrelativistic quantum theory, but it is also the top offender that impedes the integration between Feynman path integral and relativity in non-field theory. Why should this be?

For the extension, it is necessary to break up this assumption, and Eq. (1) should be written into a more general formulation in the following:

$$F(r, r\_0; t, t\_0) = R \sum\_{\text{all paths}} W(\wp) \exp\left(i S/\hbar\right) \tag{4}$$

where R is the parameter that is independent of paths and Wð Þ ℘ is the weight function with paths [13]. Additionally, some rules should be set to limit the range of choices for R and Wð Þ ℘ :

Under these four limitations, the forms of R and W pð Þ are very few. The final forms of R and W pð Þ chosen in extended Feynman path integral are

$$R = \frac{1}{\sqrt{2i\pi\hbar c^2}} \frac{H'}{\sqrt{mc^2 + H'}} ; W(\wp) = \frac{\mathbb{P}(\wp)}{\mathcal{P}(\wp)} (\Delta \tau)^{-1/2} \tag{5}$$

The H0 in Eq. (5) is the main Hamiltonian:

$$H' = \sqrt{m^2 c^4 + \left(p - A\_0\right)^2 c^2} \tag{6}$$

and

3.4. Dynamical reduction models

140 Advanced Technologies of Quantum Key Distribution

problem.

The theory of dynamical reduction models is a nonlinear and stochastic modification of the Schrödinger equation. It is proposed by Bassia and Ghirardia [10]. They integrated the master equation and linear Schrödinger equation and proposed a new nonlinear differential equation. This theory successfully solves the problems of "stochastic output" and "preferred basis" in quantum measurement and deduced the Born probability rule basing on the white noise model. However, it is still a nonrelativistic theory and remains the nonlocality

4. The extended Feynman path integral and quantum measurement

As we know, in the history of the quantum theory, there are three equivalent expressions, namely, the differential equation of Schrödinger, the matrix algebra of Heisenberg and the path integral formulation of Feynman. However, these three expressions have their own focuses. The Schrodinger and Heisenberg expressions focus on the evolution of states and operations, respectively, whereas the path integral formulation of Feynman on the "correlation" of point to point as states is evolving [11]. On the other hand, in quantum mechanics, when do a measurement on a wave function diffusing in all of space, such as the measurement of the position of an electron in the experiment of double-slit interference, we will find that the whole wave function will instantaneously collapse to this position measured with some probabilities. Obviously there may be some inner "correlation" in wave function transferring the action of the measurement from local part to whole. These two "correlations" have common

Moreover, we notice that the action integral in Feynman path integral formulation is the classical form. The classical physics is born to be a local theory and of course cannot exhibit the character of nonlocality. However, the relativity theory is different. In relativity theory, the time and space are coupling. Beyond the light cones in Minkowski space, the space-time causality is broken, and this may cause the nonlocality. The superluminal velocities are forbidden in real world, but for a connection description of virtual paths in the path integral theory, it might be practicable. What will happen when we extend the classical action to relativistic action? Could the superluminal trajectories included in possible paths to calculate quantum amplitude in the Feynman theory cause the nonlocality? How is the relationship between "unitary evolution operation" and "quantum measurement"? These questions will be revealed

4.1. Why is it concerning with the Feynman path integral?

characters and may be unified to be one.

when we extend the Feynman path integral.

4.2. How to extend the Feynman path integral?

The formulation for Feynman path integral can be written as

$$\mathbb{P}(\mathfrak{g}\mathfrak{o}) = \int\_{t\_0}^{t} |P| d\tau; \mathcal{P}(\mathfrak{g}\mathfrak{o}) = \int\_{t\_0}^{t} \left| \sqrt{2mT} \right| d\tau \tag{7}$$

calculation in 1D space for simplification. The methods of the calculation in 2D and 3D are the same. Before this calculation, we define two parameters as τ0 ¼ ℏ= mc2 � � and ε0 ¼ ε=τ0:

> 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 � � p 1=2

> > 0 @

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p

s

X m i pε ℏ

2m! du !exp ð Þ ipx

0 u �1

Figure 1. Contour integral. This figure shows the contour integral in a complex plane. The black line in figure denotes the

this contour integral, there is no singular point, and of course the total integral value is zero. Therefore,

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2 c2

2 1 � ð Þ 1 � u 2 � �m

ε0

1

� �2m 1 � u2 � �m

exp ð Þ iuε0 � iε0 du � �

0 ⋯du. The integral on the red line is always zero when j j!z ∞. For

ε0exp �i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Stochastic Quantum Potential Noise and Quantum Measurement

Aexp ð Þ �ipvε=ℏ dv

1 � v2=c2 p ε0 � �ψð Þ r0; t0 dv

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1

CCA

exp ð Þ ipx=ℏ

(12)

143

ε 1=2 0

r ε0exp �i

0 exp ð Þ �iuε0

φpexp ð Þ ipx dp ð1

I0 ¼ ðct �ct

> ¼ ð∞ �∞

> ¼ ð∞ �∞

¼ X

integral � Ð 1þi∞

Ð 1þi∞ 1 ⋯du ¼ Ð ∞

1 ⋯du � Ð 1

1 ⋯du:

⋯dr0 ¼ ε

φpdp 2Rτ

φpdp 2Rτ

m2R i pc ℏ

BB@

0

ðc �ct

> 1=2 0 ðc 0

1=2 0 ð1 0

� �2m cε2mþ1=2

2m!

Similarly, we can also get the expression of I0:

⋯dv ¼ 2Rτ

1 þ

ε 1=2 0

ð Þ 1 � u �1=2

ð∞ �∞

The contour integral is used in the last step as shown in Figure 1.

0 ⋯du; the blue line denotes Ð ∞

1=2 0 ðct �ct

ffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2 c2 � � q 1=2

> ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2 c2

> > ε 1=2

P, T and Δτ are called the momentum, kinetic energy and proper time in terms of fourdimensional space–time, respectively:

$$|P| = \frac{mv}{\sqrt{1 - v^2/c^2}},\\ T = \frac{mc^2}{\sqrt{1 - v^2/c^2}} - mc^2,\\ \Delta\tau = \int\_0^t \frac{1}{\sqrt{1 - v^2/c^2}}d\tau \tag{8}$$

The expressions of Wð Þ ℘ and R are very interesting. As we can see, under the low-energy and low-velocity condition, H0 ≪ mc2 and v ≪ c, then R ¼ 1 ffiffiffiffiffiffiffiffiffiffi 2iπℏc2 p and Wð Þ¼ ℘ ð Þ t � t0 1=2 because j j P ¼ ffiffiffiffiffiffiffiffiffi 2mT p in classical physical theory. This means Eq. (4) can be transformed into the Feynman path integral if we choose the formulations of W pð Þ and R as shown in Eq. (5). What is concerning then for us is what we can get from Eq. (4) under very high energy and velocity.

#### 4.3. The new differential equation and Klein-Gordon equation

It is hard to directly calculate the value of Eq. (4) because the path integral is not normal integral term and the normal integral method is invalid for Eq. (4). A way to get some results from Eq. (4) is to follow the method that Feynman used [11, 12]. We consider a minimal evolution time process, t ¼ t0 þ ε, where ε ! 0. In this process:

$$
\psi(r, t\_0 + \varepsilon) = \int\_{-\infty}^{\infty} \psi(r\_0, t\_0) F(r, r\_0; t\_0 + \varepsilon, t\_0) dr\_0 = R \int\_{-\infty}^{\infty} \psi(r\_0, t\_0) W(\mathfrak{g}) dr\_0 \tag{9}
$$

When ε ! 0, the weight function Wð Þ ℘ can be simply expressed the term of

$$W(\wp) = \frac{\left(1 + \sqrt{1 - v^2/c^2}\right)^{1/2}}{\varepsilon^{1/2}\sqrt{1 - v^2/c^2}}\tag{10}$$

where v ¼ ð Þ r � r0 =ε. This value can be greater than the superluminal velocity, and F rð Þ ;r0; t0 þ ε; t0 therefore will become the complex function when v > c. The integral form should be departed into two parts: the part that contains the low-velocity paths and the part that contains superluminal-velocity paths:

$$I = \int\_{-\infty}^{\infty} \psi(r\_0, t\_0) F(r, r\_0; t\_0 + \varepsilon, t\_0) dr\_0 = \int\_{-ct}^{ct} \cdots dr\_0 + \left(\int\_{ct}^{\infty} \cdots dr\_0 + \int\_{-\infty}^{-ct} \cdots dr\_0\right) = I\_0 + I\_1 \tag{11}$$

This can be exactly calculated. The amazing thing is the final result calculated for I that contains the term ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ iℏ∇ þ A0 2 c2 q . In the following context, we will detail this calculation in 1D space for simplification. The methods of the calculation in 2D and 3D are the same. Before this calculation, we define two parameters as τ0 ¼ ℏ= mc2 � � and ε0 ¼ ε=τ0:

I0 ¼ ðct �ct ⋯dr0 ¼ ε ðc �ct ⋯dv ¼ 2Rτ 1=2 0 ðct �ct 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 � � p 1=2 ε 1=2 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p ε0exp �i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p ε0 � �ψð Þ r0; t0 dv ¼ ð∞ �∞ φpdp 2Rτ 1=2 0 ðc 0 1 þ ffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2 c2 � � q 1=2 ε 1=2 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2 c2 r ε0exp �i ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2 c2 s ε0 0 @ 1 Aexp ð Þ �ipvε=ℏ dv 0 BB@ 1 CCA exp ð Þ ipx=ℏ ¼ ð∞ �∞ φpdp 2Rτ 1=2 0 ð1 0 ð Þ 1 � u �1=2 ε 1=2 0 exp ð Þ �iuε0 X m i pε ℏ � �2m 1 � u2 � �m 2m! du !exp ð Þ ipx ¼ X m2R i pc ℏ � �2m cε2mþ1=2 2m! ð∞ �∞ φpexp ð Þ ipx dp ð1 0 u �1 2 1 � ð Þ 1 � u 2 � �m exp ð Þ iuε0 � iε0 du � � (12)

Similarly, we can also get the expression of I0:

Pð Þ¼ ℘

1 � v2=c2 p , T ¼ mc2

4.3. The new differential equation and Klein-Gordon equation

evolution time process, t ¼ t0 þ ε, where ε ! 0. In this process:

ð∞ �∞

dimensional space–time, respectively:

142 Advanced Technologies of Quantum Key Distribution

and low-velocity condition, H0

ψð Þ¼ r; t0 þ ε

that contains superluminal-velocity paths:

q

ψð Þ r0; t0 F rð Þ ;r0; t0 þ ε; t0 dr0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ iℏ∇ þ A0

because j j P ¼ ffiffiffiffiffiffiffiffiffi

velocity.

I ¼ ð∞ �∞

contains the term

j j P ¼ mv ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðt t0

j j P dτ;Pð Þ¼ ℘

P, T and Δτ are called the momentum, kinetic energy and proper time in terms of four-

The expressions of Wð Þ ℘ and R are very interesting. As we can see, under the low-energy

the Feynman path integral if we choose the formulations of W pð Þ and R as shown in Eq. (5). What is concerning then for us is what we can get from Eq. (4) under very high energy and

It is hard to directly calculate the value of Eq. (4) because the path integral is not normal integral term and the normal integral method is invalid for Eq. (4). A way to get some results from Eq. (4) is to follow the method that Feynman used [11, 12].  This value can be greater than the superluminal velocity, and F rð Þ ;r0; t0 þ ε; t0 therefore will become the complex function when v > c. The integral form should be departed into two parts: the part that contains the low-velocity paths and the part

⋯dr0 þ

This can be exactly calculated. The amazing thing is the final result calculated for I that

ð∞ ct

⋯dr0 þ

ð�ct �∞

. In the following context, we will detail this

� �

⋯dr0

¼ I0 þ I1 (11)

ðct �ct

> 2 c2

ε1=2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

When ε ! 0, the weight function Wð Þ ℘ can be simply expressed the term of

Wð Þ¼ ℘

≪ mc2 and v ≪ c, then R ¼ 1 ffiffiffiffiffiffiffiffiffiffi

2mT p in classical physical theory. This means Eq. (4) can be transformed into

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p � mc2

ðt t0

ffiffiffiffiffiffiffiffiffi 2mT � � p �

, Δτ ¼

� � �

ðt t0

ð∞ �∞

1 � v2=c2 p (10)

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dτ (7)

1 � v2=c2 p dτ (8)

1=2

2iπℏc2 p and Wð Þ¼ ℘ ð Þ t � t0

ψð Þ r0; t0 Wð Þ ℘ dr0 (9)

The contour integral is used in the last step as shown in Figure 1.

Figure 1. Contour integral. This figure shows the contour integral in a complex plane. The black line in figure denotes the integral � Ð 1þi∞ 1 ⋯du � Ð 1 0 ⋯du; the blue line denotes Ð ∞ 0 ⋯du. The integral on the red line is always zero when j j!z ∞. For this contour integral, there is no singular point, and of course the total integral value is zero. Therefore, Ð 1þi∞ 1 ⋯du ¼ Ð ∞ 1 ⋯du:

$$\begin{split} I\_{0} &= \int\_{t}^{\pi} \cdots dr\_{0} + \varepsilon \int\_{-\pi}^{-\varepsilon t} \cdots dr\_{0} = 2R\tau\_{0}^{1/2} \int\_{t}^{\pi} \frac{\left(1 + \sqrt{1 - v^{2}/c^{2}}\right)^{1/2}}{\varepsilon\_{0}^{1/2} \sqrt{1 - v^{2}/c^{2}}} \cdot \varepsilon\_{0} \exp\left(-i\sqrt{1 - v^{2}/c^{2}}\varepsilon\_{0}\right) \\ &\times \left(\psi(r\_{0}, t\_{0}) + \psi(-r\_{0}, t\_{0})\right) du = \sum\_{m} 2R\left(i\frac{\nu\varepsilon}{\hbar}\right)^{2m} \frac{c\varepsilon^{2m + 1/2}}{2m!} \int\_{-\infty}^{\infty} \varphi\_{p} \exp\left(i\mathrm{p}x\right) \\ &\times dp \left(\int\_{1}^{1+i\nu} u^{\frac{\pi}{2}} \left(1 - \left(1 - u\right)^{2}\right)^{m} \exp\left(i\nu\varepsilon\_{0} - i\varepsilon\_{0}\right) du\right) \\ &= \sum\_{m} 2R\left(i\frac{\nu\varepsilon}{\hbar}\right)^{2m} \frac{c\varepsilon^{2m + 1/2}}{2m!} \int\_{-\infty}^{\infty} \varphi\_{p} \exp\left(i\mathrm{p}x\right) dp \left(\int\_{1}^{\nu} u^{\frac{\pi}{2}} \left(1 - \left(1 - u\right)^{2}\right)^{m} \exp\left(i\nu\varepsilon\_{0} - i\varepsilon\_{0}\right) du\right) \end{split} \tag{13}$$

iℏ d

dtψð Þ¼ r; t

iℏ d

iℏ d

iℏ d

interpretation for Klein-Cordon equation by EFPI theory.

4.4. The extended Feynman path integral and density-flux equation

equation.

Combining Eqs. (20) and (21), we get these two equations:

dt � V rð Þ � �ϕ� ¼ �

dt � V rð Þ � �ψþ ¼ �

dt � V rð Þ � �ψ� ¼ �

The more general formulation in 3D is

iℏ d

dtψð Þ¼ x; t

q

deduction can be seen in supplementary online material of the reference [13].

function ϕþ. ϕþ satisfied the relation that Eq. (20) has shown and ϕ� is satisfied

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ icℏ∂x 2

> 2 c2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ iℏ∇ � A0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ iℏ∇ � A0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ iℏ∇ � A0

2 c2

2 c2

2 c2

þ V rð Þ � �ψð Þ r; t (20)

Stochastic Quantum Potential Noise and Quantum Measurement

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ iℏ∇ � A0

It is more complicated to get Eq. (20), and we will not detail it in this chapter. The detailed

It should be mentioned that Eq. (20) is not a covariant equation under the Lorentz transformation. To construct a Lorentz covariant, the antiparticle wave function should be introduced. The antiparticle wave function is denoted as ϕ� to be distinguished from the particle wave

q

q

q

where and . Eqs. (22) and (23) are the Klein-Gordon

In 1926, Oskar Klein and Walter Gordon proposed this relativistic wave equation. However, it was found later that this equation is not suitable for one particle because the probability density is not a positive quantity, which means the particle can be created and annihilated arbitrarily in Klein-Gordon equation [14]. The extended Feynman path integral shows the explanation for this non-positive probability density here. The wave function that is determined by Klein-Gordon equation is the mixed state of the particle and its antiparticle. Because particles and antiparticles can be annihilated each other to a vacuum state, and the vacuum state can produce particles and antiparticles, so the mixed state with superposition state of a particle and an antiparticle is a matter of course of a non-positive quantity. This is the physical

In quantum mechanics, the continuity equation describes the conservation of probability density in the transport process. It is a local form of conservation laws. It says the probability cannot be created or annihilated and, at the same time, also cannot be teleported from one

ψð Þ x; t (19)

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145

ϕ� (21)

ψþ (22)

ψ� (23)

Integrating Eq. (12) and Eq. (13), we get the conclusion finally:

$$\begin{split} I &= \sum\_{m} 2R\left(l\frac{\mu}{\hbar}\right)^{2m} \frac{c\mathcal{E}^{2m+1/2}}{2m!} \int\_{0}^{\infty} \wp\_{p} \exp\left(ip\mathbf{x}\right) dp \left(\int\_{0}^{u} u^{\frac{1}{2}} \left(1 - \left(1 - u\right)^{2}\right)^{m} \exp\left(i\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}\_{0} - i\boldsymbol{\varepsilon}\_{0}\right) du\right) \\ &= \int\_{0}^{\infty} \sum\_{m} 2R\left(l\frac{\mu}{\hbar}\right)^{2m} \frac{c\mathcal{E}^{2m + \frac{1}{2}}}{2m!} \Gamma\left(2m + \frac{1}{2}\right) M\left(-m, \frac{1}{2} - 2m, -2i\boldsymbol{\varepsilon}\_{0}\right) \wp\_{p} \exp\left(i\boldsymbol{p}\boldsymbol{x}\right) dp \end{split} \tag{14}$$

The function M að Þ ; b; z is the Kummer's function (confluent hypergeometric function) and equals

$$M\left(-m, \frac{1}{2} - 2m, -2i\varepsilon\_0\right) = \sum\_{n} \frac{m!}{n!} \frac{(4m-1)!}{n!} \left(-i\varepsilon\_0\right)^n \tag{15}$$

Summation in Eq. (14) is then

$$\sum\_{m} \mathcal{R}\left(i\frac{pc}{\hbar}\right)^{2m} \frac{c\varepsilon^{2m+\frac{1}{2}}}{2m!} \Gamma\left(2m+\frac{1}{2}\right) M\left(-m, \frac{1}{2} - 2m, -2i\varepsilon\_0\right) = \exp\left(\frac{-i\sqrt{m^2c^4 + p^2c^2}\varepsilon}{\hbar}\right) \tag{16}$$

And Eq. (14) can be further simplified:

$$I = \int\_0^\infty \exp\left(\frac{-i\sqrt{m^2c^4 + p^2c^2}\varepsilon}{\hbar}\right) \varphi\_p \exp\left(ipx\right) dp$$

$$\exp\left(\frac{-i\sqrt{m^2c^4 + (-ic\hbar\nabla\_x)\varepsilon}}{\hbar}\right) \int\_0^\infty \varphi\_p \exp(ipx) dp = \exp\left(\frac{-i\sqrt{m^2c^4 + (-ic\hbar\partial\_x)\varepsilon}}{\hbar}\right) \psi(x, t\_0)$$

It is, namely:

$$\psi(\mathbf{x}, t\_0 + \varepsilon) = \exp\left(\frac{-i\sqrt{m^2c^4 + (-ic\hbar\mathfrak{d}\_\mathbf{x})^2}\varepsilon}{\hbar}\right)\psi(\mathbf{x}, t\_0) \tag{18}$$

Hence, the new differential equation we get in this extended Feynman path integral is

Stochastic Quantum Potential Noise and Quantum Measurement http://dx.doi.org/10.5772/intechopen.74253 145

$$i\hbar\frac{d}{dt}\psi(\mathbf{x},t) = \sqrt{m^2c^4 + \left(-ic\hbar\partial\_x\right)^2}\psi(\mathbf{x},t)\tag{19}$$

The more general formulation in 3D is

I0 ¼ Ð ∞

ct ⋯dr0 þ ε

144 Advanced Technologies of Quantum Key Distribution

m2R i pc ℏ

m2R i pc ℏ

> m2R i pc ℏ � �2m cε2mþ1

Summation in Eq. (14) is then

� �2m cε2mþ1

And Eq.  (12) and Eq. (13), we get the conclusion finally:

ð∞ 0

2 2m! Γ 2m þ

1

2 � 2m; �2iε0 � �

> 1 2 � �

ψð Þ¼ x; t0 þ ε exp �i

M �m;

0 exp �i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

m2c4 þ p2c2 p ε ℏ !

q

Hence, the new differential equation we get in this extended Feynman path integral is

0 @

M �m;

2 2m! Γ 2m þ

I ¼ Ð ∞

1=2 0 Ð ∞ ct

> m2R i pc ℏ

φpexp ð Þ ipx dp

φpexp ð Þ ipx dp

1 2 � �

1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 � � p 1=2

� �2m cε2mþ1=2 2m!

> ð∞ 1 u �1

ð∞ 0 u �1

1

M �m;

The function M að Þ ; b; z is the Kummer's function (confluent hypergeometric function) and

¼ X n m! n!

2 � 2m; �2iε0 � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p

> ð∞ �∞

2 1 � ð Þ 1 � u 2 � �m

2 1 � ð Þ 1 � u 2 � �m

ð Þ 4m � 1 !

φpexp ð Þ ipx dp

ε

1

0 φpexp ð Þ ipx dp ¼ exp �i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ icℏ∂x 2

ℏ

2 � 2m; �2iε0 � �

ε0exp �i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

φpexp ð Þ ipx

� �

φpexp ð Þ ipx dp

¼ exp �i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m2c4 þ �ð Þ icℏ∂x p ε ℏ !

� �

1 � v2=c2 p ε0 � �

exp ð Þ iuε0 � iε0 du

exp ð Þ iuε0 � iε0 du

n! ð Þ �iε0 n (15)

m2c4 þ p2c2 p ε ℏ !

ψð Þ x; t0

Aψð Þ x; t0 (18)

(13)

(14)

(16)

(17)

ε 1=2 0

exp ð Þ iuε0 � iε0 du

$$i\hbar\frac{d}{dt}\psi(r,t) = \left(\sqrt{m^2c^4 + \left(-i\hbar\nabla - A\_0\right)^2c^2} + V(r)\right)\psi(r,t)\tag{20}$$

It is more complicated to get Eq. (20), and we will not detail it in this chapter. The detailed deduction can be seen in supplementary online material of the reference [13].

It should be mentioned that Eq. (20) is not a covariant equation under the Lorentz transformation. To construct a Lorentz covariant, the antiparticle wave function should be introduced. The antiparticle wave function is denoted as ϕ� to be distinguished from the particle wave function ϕþ. ϕþ satisfied the relation that Eq. (20) has shown and ϕ� is satisfied

$$\left(i\hbar\frac{d}{dt} - V(r)\right)\phi\_- = -\sqrt{m^2c^4 + \left(-i\hbar\nabla - A\_0\right)^2c^2}\phi\_-\tag{21}$$

Combining Eqs. (20) and (21), we get these two equations:

$$\left(i\hbar\frac{d}{dt} - V(r)\right)\psi\_{+} = -\sqrt{m^{2}c^{4} + \left(-i\hbar\nabla - A\_{0}\right)^{2}c^{2}}\psi\_{+}\tag{22}$$

$$\left(i\hbar\frac{d}{dt} - V(r)\right)\psi\_- = -\sqrt{m^2c^4 + \left(-i\hbar\nabla - A\_0\right)^2c^2}\psi\_-\tag{23}$$

where and . Eqs. (22) and (23) are the Klein-Gordon equation.

In 1926, Oskar Klein and Walter Gordon proposed this relativistic wave equation. However, it was found later that this equation is not suitable for one particle because the probability density is not a positive quantity, which means the particle can be created and annihilated arbitrarily in Klein-Gordon equation [14]. The extended Feynman path integral shows the explanation for this non-positive probability density here. The wave function that is determined by Klein-Gordon equation is the mixed state of the particle and its antiparticle. Because particles and antiparticles can be annihilated each other to a vacuum state, and the vacuum state can produce particles and antiparticles, so the mixed state with superposition state of a particle and an antiparticle is a matter of course of a non-positive quantity. This is the physical interpretation for Klein-Cordon equation by EFPI theory.

#### 4.4. The extended Feynman path integral and density-flux equation

In quantum mechanics, the continuity equation describes the conservation of probability density in the transport process. It is a local form of conservation laws. It says the probability cannot be created or annihilated and, at the same time, also cannot be teleported from one place to another. However, in the extended Feynman path integral, the density-flux equation will be revised, and the local conservation is broken.

In extended Feynman path integral, the density-flux equation can be written as the following formula:

$$\frac{\partial \rho(r,t)}{\partial t} + \nabla \cdot \dot{\boldsymbol{j}} + \sum\_{n=2}^{\text{res}} \boldsymbol{B}\_{n} \nabla^{n} \cdot \boldsymbol{Q}\_{n}(r,t) = \mathbf{0} \tag{24}$$

Considering a minimum time-evolution process, the propagator is

A0ðx0, tÞd0 exists in the integral formula of Eq. (28), then limε!0

Stochastic Quantum Potential Noise and Quantum Measurement

http://dx.doi.org/10.5772/intechopen.74253

F xð Þ 1; x0; t0 þ ε; t0 6¼ δð Þ x1 � x0 . This is different from the normal propagator K xð Þ ; x0; t0 þ ε; t0 shown in Eq. (2), because limε!0 K xð ; x0; t0 þ ε; t0Þ ¼ δð Þ x � x0 . This difference, caused by relativistic effect of paths, is the root that produces the nonlocality in quantum measurement

limε!0 F xð Þ 1; x0; t0 þ ε; t0 6¼ δð Þ x0 � x0 means the change of arbitrary point should spend time to propagate the other point and exhibit stronge nonlocal space-time character. If the value of wave function at x ¼ x0 changes, the whole wave function will change for the nonlocal prop-

Rb ≈Rb0 1 � AIc2ð Þ bp � A0

After this definition, we will show how the measurement happens under the potential noise.

H0, if we put the potential noise in this system, the initial state will change. We denote the evolution state in arbitrary time t as ψðx, tÞ. The ψðx, tÞ can be expanded with basis states

H0

P

mamφm. The task for us is to find out the varying value of am under each

� � (30)

mamφmð Þx , where φm is the eigenstate of

Because the term Ð x

process. In fact:

Therefore

We define Rb0 ¼ 1 ffiffiffiffiffiffiffiffiffiffi

φm as ψðx, tÞ ¼ P

perturbational noise:

2iπℏc2 p

x�η

agator. In the followings, we will detail this character.

H0 ffiffiffiffiffiffiffiffiffiffiffiffi mc2þH0 p ; then

Considering an initial state with the form ψð Þ¼ x; t0

ð28Þ

147

ð29Þ

where Qnð Þ¼ r; t ψ∗∇nψ � ψ∇nψ∗ and Bn ¼��ð Þ iℏ 2n�1 c2n= mc2 � �2n�1 . The last term in the right of Eq. (24) is caused by relativistic effect and breaks the local conservation.

#### 4.5. The wave function collapse in extended Feynman path integral

From the theory of Neumann, the difficulties of understanding collapse are the probability, which seems incompatible with the deterministic time-evolution equation, and the instantaneity, which seems that it breaks the special relativity theory. In this section, we will show that these puzzling characters are due to the potential noise and nonlocal correlation (or relativistic effect).

Let us return to Eq. (9). The superluminal paths are included when we calculate the propagator. The superluminal paths will support complex phases in Eq. (9), and these phases cannot be canceled by each other like the real phases in Feynman path integral theory. These complex phases are the main culprits that cause the nonlocal correlation.

To describe this mechanism concisely, the nonlocal correlation produced in 1D space is just detailed here. Assume a system in the potential field with the scalar potential U xð Þ and vector A0ð Þx . A potential noise AIð Þt is under this system and satisfies the white noise equations, namely:

$$
\langle A\_I(t\_1) A\_I(t\_0) \rangle = \frac{2mk\_bT}{\eta} \delta(t\_1 - t\_0); \langle A\_I(t) \rangle = 0 \tag{25}
$$

The Hamiltonian of this system is then

$$H = \sqrt{m^2 c^4 + \left(-i\hbar \partial\_x - (A\_0 + A\_I)\right)^2 c^2} + V(\mathbf{x}) \tag{26}$$

And we define a new Hamiltonian without potential noise as

$$H\_0 = \sqrt{m^2c^4 + \left(-i\hbar\partial\_\mathbf{x} - A\_0\right)^2c^2} + V(\mathbf{x})\tag{27}$$

We will see later that H0 is very important in quantum measurement, because it determines the basis-state-space that the wave function collapses into. The basis-preferred problem puzzles us for many years; we do not know why the system measured prefers to collapse into some set of basis state. According to the extended Feynman path integral theory, the preferred basis is depended by the Hamiltonian H0. This will be detailed in the following.

Considering a minimum time-evolution process, the propagator is

$$F(\mathbf{x}\_1, \mathbf{x}\_0; t\_0 + \varepsilon, \mathbf{t}\_0) = \mathcal{R} \sqrt{\frac{c}{l\eta}} \exp(-mc|\eta|\hbar^{-1} + i\hbar^{-1} \int\_{\mathbf{x}-\eta}^{\mathbf{x}} A\_0(\mathbf{x}\_0, t) d\mathbf{x}\_0) \tag{28}$$

Because the term Ð x x�η A0ðx0, tÞd0 exists in the integral formula of Eq. (28), then limε!0 F xð Þ 1; x0; t0 þ ε; t0 6¼ δð Þ x1 � x0 . This is different from the normal propagator K xð Þ ; x0; t0 þ ε; t0 shown in Eq. (2), because limε!0 K xð ; x0; t0 þ ε; t0Þ ¼ δð Þ x � x0 . This difference, caused by relativistic effect of paths, is the root that produces the nonlocality in quantum measurement process.

In fact:

place to another. However, in the extended Feynman path integral, the density-flux equation

In extended Feynman path integral, the density-flux equation can be written as the following

n¼2

From the theory of Neumann, the difficulties of understanding collapse are the probability, which seems incompatible with the deterministic time-evolution equation, and the instantaneity, which seems that it breaks the special relativity theory. In this section, we will show that these puzzling characters are due to the potential noise and nonlocal correlation (or relativistic

Let us return to Eq. (9). The superluminal paths are included when we calculate the propagator. The superluminal paths will support complex phases in Eq. (9), and these phases cannot be canceled by each other like the real phases in Feynman path integral theory. These complex

To describe this mechanism concisely, the nonlocal correlation produced in 1D space is just detailed here. Assume a system in the potential field with the scalar potential U xð Þ and vector A0ð Þx . A potential noise AIð Þt is under this system and satisfies the white noise equations,

η

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ iℏ∂x � ð Þ A0 þ AI

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2c4 þ �ð Þ iℏ∂x � A0

We will see later that H0 is very important in quantum measurement, because it determines the basis-state-space that the wave function collapses into. The basis-preferred problem puzzles us for many years; we do not know why the system measured prefers to collapse into some set of basis state. According to the extended Feynman path integral theory, the preferred basis is

2 c2

2 c2

Bn∇n � Qnð Þ¼ r; t 0 (24)

δð Þ t1 � t0 ; Ah i Ið Þt ¼ 0 (25)

þ V xð Þ (26)

þ V xð Þ (27)

. The last term in the right

c2n= mc2 � �2n�1

will be revised, and the local conservation is broken.

146 Advanced Technologies of Quantum Key Distribution

∂rð Þ r; t

where Qnð Þ¼ r; t ψ∗∇nψ � ψ∇nψ∗ and Bn ¼��ð Þ iℏ 2n�1

∂t þ ∇ � j þX∞

of Eq. (24) is caused by relativistic effect and breaks the local conservation.

4.5. The wave function collapse in extended Feynman path integral

phases are the main culprits that cause the nonlocal correlation.

The Hamiltonian of this system is then

H ¼

q

And we define a new Hamiltonian without potential noise as

H0 ¼

q

depended by the Hamiltonian H0. This will be detailed in the following.

h i AIð Þ t1 AIð Þ t0 ¼ 2mkbT

formula:

effect).

namely:

$$\int\_{t\_0}^{t\_0+\varepsilon} -mc^2\sqrt{1-v^2/c^2} \, dt = \int\_{t\_0}^{t\_0+\varepsilon} -mc^2\sqrt{(dt)^2 - dx^2/c^2} = imc\{\Delta x\};$$

$$\int\_{t\_0}^{t\_0+\varepsilon} (-U(\varkappa) + A\nu)dt = A\Delta\varkappa$$

Therefore

$$F(\mathbf{x}\_1, \mathbf{x}\_0; \mathbf{t}\_0 + \varepsilon, \mathbf{t}\_0) = \frac{1}{\sqrt{2\pi\hbar c^2}} \frac{\hbar^{\prime}}{\sqrt{mc^2 + \hbar^{\prime}}} \sqrt{\frac{c}{\hbar}} \exp\{-mc|\eta|\hbar^{-1} + i\hbar^{-1} \int\_{\mathbf{x} - \eta}^{\mathbf{x}} A\_0(\mathbf{x}\_0, \mathbf{t}) d\mathbf{x}\_0\} \tag{29}$$

$$\psi\_- = 1/\sqrt{2} (\phi\_+ - \phi\_-)$$

limε!0 F xð Þ 1; x0; t0 þ ε; t0 6¼ δð Þ x0 � x0 means the change of arbitrary point should spend time to propagate the other point and exhibit stronge nonlocal space-time character. If the value of wave function at x ¼ x0 changes, the whole wave function will change for the nonlocal propagator. In the followings, we will detail this character.

We define Rb0 ¼ 1 ffiffiffiffiffiffiffiffiffiffi 2iπℏc2 p H0 ffiffiffiffiffiffiffiffiffiffiffiffi mc2þH0 p ; then

$$
\widehat{R} \approx \widehat{R}\_0 \left( 1 - \frac{A\_I c^2 (\widehat{p} - A\_0)}{H\_0} \right) \tag{30}
$$

After this definition, we will show how the measurement happens under the potential noise. Considering an initial state with the form ψð Þ¼ x; t0 P mamφmð Þx , where φm is the eigenstate of H0, if we put the potential noise in this system, the initial state will change. We denote the evolution state in arbitrary time t as ψðx, tÞ. The ψðx, tÞ can be expanded with basis states φm as ψðx, tÞ ¼ P mamφm. The task for us is to find out the varying value of am under each perturbational noise:

$$\begin{split} a\_n(t+\delta) &= \int\_{-\infty}^{\infty} \wp\_n(u,t) \* \phi(u,t) du \\ &= \int\_{-\infty}^{\infty} \wp\_n(u,t) \* c^{\frac{1}{2}} \Re(u,t) \* \int\_{-\infty}^{\infty} (i\eta)^{-\frac{1}{2}} \exp\left(-\frac{mc|\eta|}{\hbar}\right) \exp(\xi\_l)\phi(u-\eta,t) \,d\eta \\ &= \sum\_m a\_m(t)\,\lambda\_{n,m}(t-\delta) \left(1 + \frac{A\_lc^2 p\_n}{E\_n}\delta\_{n,m}\right) \end{split}$$

After rearranging the equation above, we get

$$a\_n(t+\delta) = \sum\_m a\_m(t)D\_{m,n}(t-\delta)\tag{31}$$

attentions of physicists since the beginning of the quantum theory establishment, but there is still no consensus. The measurement problem blocks up the way for us to understand the nonlocality and manipulate quantum state. Can the quantum measurement be controlled? Can we get the definite output we want under every measurement? If the quantum measurement can be controlled, the teleportation without classical communication channel can be realized, and the aim of superfast manipulation for quantum state will arrive. We can even transfer the energy thought nonlocality under controlled quantum measurement and make more novel encryption scheme for quantum communication. However, the key problem is "can we control

Figure 2. The process of collapse under a "potential noise". (a) The red line denotes the absolute value of probability

line is the function of potential. The different sets of noise cause the different collapse results. According the simulation, the process time of collapse is 0.3 ns in the top picture and 0.1 ns in the bottom picture. (b) The function of AI shown in

amplitude a0ð Þt with the initial value 1/2, and the blue one denotes a1ð Þt with the initial value ffiffiffi

The extended Feynman path integral mechanism answered this question. According to this mechanism, the character, "stochastic output" and "instantaneous collapse process" of quantum measurement are rooted in the "random" potential noise and "nonlocal" wave function inner correlation. The "nonlocality" is caused by the "relativistic effect" of superluminal paths in path integral theory. The superluminal paths will support a complex action function S in

theory cannot be canceled and makes F xð Þ 1; x0; t0; t0 6¼ δð Þ x1 � x0 . This relation reveals that the propagator is no longer a local correlation. All points in space are correlated simultaneously, and any local perturbation will simultaneously transfer into the whole space. The extended Feynman path integral gives a simulation for two-energy-level system and exhibits that the

1 � v2=c2 p of S. This complex action that acted as a phase in integral

3

Stochastic Quantum Potential Noise and Quantum Measurement

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149

p =2. The black oscillatory

the quantum measurement?" If yes, how? If no, why?

Eq. (4) for the expression ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eq. (32).

where

$$D\_{m,n}(t-\delta) = \lambda\_{n,m}(t-\delta)\left(1 + \frac{A\_I c^2 p\_n}{E\_n} \delta\_{n,m}\right)$$

$$\lambda\_{n,m}(t-\delta) = \int\_{-\infty}^{+\infty} \varphi\_n(\mathbf{x}) \mathbf{R}(\mathbf{x}, t-\delta) \* \widehat{R}\_0^{-1} \varphi\_m(\mathbf{x}) d\mathbf{x}$$

$$\mathbf{R}(\mathbf{x}, t) = \frac{\psi(\mathbf{x}, t)}{\widehat{R}\_0^{-1} \psi(\mathbf{x}, t)}$$

δ is the time interval of the neighbor potential noise pulses. In fact, to simulate the process of quantum measurement under potential noise, we let

$$A\_{I} = \sum\_{n=0}^{\infty} \left(\frac{2mk\_{b}T}{\eta \Delta}\right)^{1/2} \text{Random}(n) \left(\Theta(t - n\delta) - \Theta(t - (n-1)\delta)\right) \tag{32}$$

We simulate the collapse process of a wave function with the form j i ψ ¼ 1=2 0j i þ ffiffiffi 3 p =2 1j i, where 0j i and 1j i are the harmonic-oscillator basis. According the simulation, we show the j i ψ will randomly collapse into 0j i or 1j i quickly (Figure 2).

#### 5. Conclusions

Measurement, in quantum theory, is not just a theory concerning the Schrödinger cat that is alive or dead, or the moon being here or not, but also the key and basis to the problem of the interpretation of quantum mechanics. In fact, the different views for the quantum measurement yield different interpretation for quantum mechanics, such as the Copenhagen interpretation, relative-state interpretation, Bohmian mechanics and so on. It has attracted many

After rearranging the equation above, we get

148 Advanced Technologies of Quantum Key Distribution

where

anð Þ¼ t þ δ X

Dm,nð Þ¼ t � δ λn,mð Þ t � δ 1 þ

ðþ∞ �∞

λn,mð Þ¼ t � δ

quantum measurement under potential noise, we let

n¼0

will randomly collapse into 0j i or 1j i quickly (Figure 2).

2mkbT ηΔ � �1=2

AI ¼ X∞

5. Conclusions

mamð Þt Dm,nð Þ t � δ (31)

AIc2pn En

φnð Þx R xð Þ ; t � δ ∗Rb�1

R xð Þ¼ ; t ψð Þ x; t Rb�1 0 ψð Þ x; t

δ is the time interval of the neighbor potential noise pulses. In fact, to simulate the process of

We simulate the collapse process of a wave function with the form j i ψ ¼ 1=2 0j i þ ffiffiffi

where 0j i and 1j i are the harmonic-oscillator basis. According the simulation, we show the j i ψ

Measurement, in quantum theory, is not just a theory concerning the Schrödinger cat that is alive or dead, or the moon being here or not, but also the key and basis to the problem of the interpretation of quantum mechanics. In fact, the different views for the quantum measurement yield different interpretation for quantum mechanics, such as the Copenhagen interpretation, relative-state interpretation, Bohmian mechanics and so on. It has attracted many

� �

δn,m

0 φmð Þx dx

Random nð Þð Þ θð Þ� t � nδ θð Þ t � ð Þ n � 1 δ (32)

3 p =2 1j i,

Figure 2. The process of collapse under a "potential noise". (a) The red line denotes the absolute value of probability amplitude a0ð Þt with the initial value 1/2, and the blue one denotes a1ð Þt with the initial value ffiffiffi 3 p =2. The black oscillatory line is the function of potential. The different sets of noise cause the different collapse results. According the simulation, the process time of collapse is 0.3 ns in the top picture and 0.1 ns in the bottom picture. (b) The function of AI shown in Eq. (32).

attentions of physicists since the beginning of the quantum theory establishment, but there is still no consensus. The measurement problem blocks up the way for us to understand the nonlocality and manipulate quantum state. Can the quantum measurement be controlled? Can we get the definite output we want under every measurement? If the quantum measurement can be controlled, the teleportation without classical communication channel can be realized, and the aim of superfast manipulation for quantum state will arrive. We can even transfer the energy thought nonlocality under controlled quantum measurement and make more novel encryption scheme for quantum communication. However, the key problem is "can we control the quantum measurement?" If yes, how? If no, why?

The extended Feynman path integral mechanism answered this question. According to this mechanism, the character, "stochastic output" and "instantaneous collapse process" of quantum measurement are rooted in the "random" potential noise and "nonlocal" wave function inner correlation. The "nonlocality" is caused by the "relativistic effect" of superluminal paths in path integral theory. The superluminal paths will support a complex action function S in Eq. (4) for the expression ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � v2=c2 p of S. This complex action that acted as a phase in integral theory cannot be canceled and makes F xð Þ 1; x0; t0; t0 6¼ δð Þ x1 � x0 . This relation reveals that the propagator is no longer a local correlation. All points in space are correlated simultaneously, and any local perturbation will simultaneously transfer into the whole space. The extended Feynman path integral gives a simulation for two-energy-level system and exhibits that the potential noise can indeed lead to the collapse state randomly and rapidly. Therefore, the key to control the quantum measurement is to control the potential noise exactly. "Potential noise" is caused by thermal fluctuation of potential filed or irregularity potential boundary. How to control this potential noise is still an unsolved topic.

[4] Stapp HP. The Basis Problem in many-worlds Theories. Springer; 2007. DOI: https://doi.

Stochastic Quantum Potential Noise and Quantum Measurement

http://dx.doi.org/10.5772/intechopen.74253

151

[5] Żurek WH. Decoherence and the transition from quantum to classical. Physics Today.

[6] Żurek WH. Decoherence, einselection, and the quantum origins of the classical. Reviews

[7] Zeh HD. On the interpretation of measurement in quantum theory. Foundations of Phys-

[8] Albert D, Loewer B. Interpreting the many-worlds interpretation. Synthese. 1988;77:195-

[9] Zeh HD The problem of conscious observation in quantum mechanical description. Foun-

[10] Bassia A, Ghirardia GC. Dynamical reduction models. Physics Report. 2003;379:257-456.

[11] Feynman RP. Space-time approach to non-relativistic quantum mechanics. Reviews of Modern Physics. 1948;20:367. DOI: https://doi.org/10.1103/RevModPhys.20.367

[12] Claude Grrrod. Hamiltonian Path-Integral Methods. Reviews of Modern Physics. 1966;

[13] Wen W, Bai Y. Quantum measurement and extended Feynman path integral. Communi-

[14] Herman F, Felix V. Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles.

Reviews of Modern Physics. 1958;30(1):24. DOI: 10.1103/RevModPhys.30.24

org/10.1007/978-3-540-72414-8\_11

of Modern Physics. 2003;75:715-775

dation of Physics Letters. 2000;13:221-233

cations in Theoretical Physics. 2012;57(6)

DOI: 10.1016/S0370-1573(03)00103-0

1991;44(10):36-44

ics. 1969;1(1):69-76

213

38(3):483

The extended Feynman path integral mechanism also solves the "basis-preferred" problem in quantum measurement. It exhibits the reason that the state prefers to collapse some set of basis states, which is due to the main Hamiltonian H0 defined in Eq. (27). H0 is the Hamiltonian that contains no noise. The eigenstates are the basis state that wave function prefers to collapse into.

The extended Feynman path integral mechanism shows the relation between "quantum measurement" and "unitary evolution operation". They are one and the same thing but are departed by jumpy potential noise. In mathematics, the function of potential noise is nowhere differentiable functions, and therefore, the path integral shown in Eq. (4) is not the regular path integral function under a noised potential. This is the main difference between "quantum measurement" and "unitary evolution operation" in mathematics. In physics, each potential noise point can be quickly absorbed by wave function through the nonlocality correlation, and the amounts of noise points will quickly accumulate to be a big quantity to change the whole wave function jRbj.

Additionally, besides the potential noise, the condition that the quantum measurement happens is that the interaction of system and environment should be big enough to distinguish the preferred basis state " φn � �". If the interaction is not big enough, φnjRbjφn D E ≈ φmjRbjφm D E and then Dmn ! δm,n in Eq. (31), then the collapse will not happen. In other words, the instrument that can realize the quantum measurement should be "macro" enough to produce enough noise and have big enough energy gaps of a system measured.


**3. Snakebite envenoming**

Snake envenoming is a major health issue worldwide and often considered a rationalization for morbidity and mortality (morbi-mortality) and a variety of losses, both socially and economically. Conservative estimates of the global incidence of snakebite cases suggest that approximately 5.5 million incidents are reported each year [19]. Snakebite envenoming is a potentially fatal disease that normally occurs as a result of the injection of venom following the bite of a venomous snake. Snakebite envenoming can also be caused by the spraying of venom into a person's eyes by particular snake species that are capable of spitting venom as a defense measure. Envenoming results from an estimated 50–55% of all snakebites. Snakebite envenoming in humans and animals affects multiple organ systems based on the specific snake species and the groups of toxins present in the venom and can cause *inter alia* neuromuscular paralysis, hemorrhage and protracted disruption of hemostasis, cardiotoxicity, tissue necrosis, thrombosis, and myolysis (muscle degeneration) [1, 2].

Taxonomically, snakes are carnivorous in nature, classified as reptiles and may be elapids, vipers, colubrids, boids, or pythons. Whereas the majority of bites occur when snakes are inadvertently stepped on by barefooted or unprotected victims, others may be instigated by alcohol intoxication and malevolence. In terms of classification, more than 3500 snake species are believed to exist, with venomous species accounting for approximately 600 (15–17%) [19].

#### **3.1 Epidemiology of snakebite envenoming**

Available epidemiological data on the control of snakebite envenoming has largely been fragmented, and for that matter inaccurate, a problem attributable to inadequate control efforts or measures. Rather than seeking help from health centers or hospitals, many snakebite victims resort to traditional remedies, a practice which further reduces the accuracy of existing data. Internationally, data available on snakebite reports is generally limited. Particularly, in developing countries, nearly all snakebites and the consequent deaths go unreported. In countries with poor infrastructure in rural areas, the extent of under-reporting is believed to be in excess of 70%. In spite of this, of the estimated 4.5–5.4 million snakebites yearly, close to half (1.8–2.7 million) result in clinical illness, with about 81,000–138,000 deaths due to complications. Globally, snakebites tend to affect rural populations of low socioeconomic status disproportionately, especially with groups considered as high risk including farmers or agricultural workers, hunters, fishermen, herders, occupants of poorly developed houses, working children, and people with little or no access to healthcare and

*Perspective Chapter: Diagnostic and Antivenom Immunotherapeutic Approaches… DOI: http://dx.doi.org/10.5772/intechopen.112147*

education. High-case fatalities are more common among children, while young people are more frequently affected in terms of morbidities and mortalities. In addition, access to medical care is quite challenging for women from certain cultural backgrounds, with pregnant women being exceedingly vulnerable. Across the world, snakebite envenoming and its associated deaths are unevenly distributed, with the highest cases in Asia (South and South-East) and sub-Saharan Africa. A majority of the world's population is thought to live in these areas, a phenomenon that brings about human-snake conflicts. On the other hand, Australia, Europe, and together with North America have the lowest cases of envenoming [2, 20].

#### **3.2 Mechanism and pathogenesis of snakebite envenoming**

Snake venom is produced and stored in a dedicated gland. During a bite, the venom is released through a compressor muscle-mediated action in which muscle surrounds the venom gland. Through the help of a duct, the venom is delivered to the fangs, which is then delivered into the tissues of a victim. Subsequent to snakebite envenoming, the venom toxins may act on different parts of the body systems. The venom of snakes is delivered or injected *via* a specialized delivery system. In elapids, viperids, and lamprophiids, the system is composed of a group of fangs located anterior to the maxillary bones. However, in colubrids with no front fangs, the fangs are positioned posteriorly. The injection of snake venom may either be subcutaneous or intramuscular based on the size of the fangs. Following envenoming, certain toxic components of the venom evoke pathological effects localized within the surrounding tissues, while some other toxins exert their effects on various organs as a result of their systemic distribution through the blood vessels and lymphatic system [21]. Following bites from vipers and certain cobras, local swelling is detected in a space of 2–4 hours with the potential of extending rapidly to reach its peak 2 or 3 days later. Within 2– 12 hours of the bite, blistering develops and tissue necrosis becomes evident on the first day. In the subsequent weeks or month, the necrotic tissue undergoes sloughing while secondary infections such as osteomyelitis develops. Swelling of the bitten limb may be completely resolved, and normal function restored only after some weeks. Systemic envenoming may be indicated by syncope or vomiting occurring within minutes of the bite. A few hours after the bite, coagulopathy and bleeding set in and can persevere for two or more weeks without treatment. Signs of neurotoxicity can advance to widespread flaccid paralysis and respiratory arrest within half an hour to a few hours [22, 23]. Paralysis resulting from elapid bites including mambas, cobras, and kraits may be reversed if treated with specific antivenoms or acetylcholinesterase inhibitors, with recovery over a period of time in all instances, so long as respiration is supported sufficiently. Occasionally, drowsiness may also be observed with other local symptoms such as moderate or no pain, local swelling (mostly devoid of blisters or necrosis), and paraesthesia [22, 23], In addition, envenomings from non-front-fanged colubrids including boomslangs of African origin and vine snakes are usually typified by bruising (also called ecchymosis), which tends to evolve late or slowly, coagulopathy, systemic bleeding, and severe injury to the kidney, even when envenoming is minimal [24].

#### **3.3 Diagnosis of snakebites**

Snakebite envenoming is an emergency that presents clinical and diagnostic challenges due to the potentially rapid lethal effect. Decision-making can be complicated

by the doubts surrounding the identity of a snake species, the amount of venom injected during a bite, and the composition therein, which may vary depending on the age of the snake and intra-species within its location. Nurses and health assistants are the personnel who manage most cases of snakebites in health posts, dispensaries, clinics, as well as district and rural-level hospitals. Sometimes, it is possible to refer cases to tertiary-level facilities with laboratories, specialists, and intensive care units [1].

#### *3.3.1 Clinical diagnosis*

The clinical diagnosis of snakebite envenoming relies on identifying specific signs in the patient. These include local signs such as swelling, blistering, and local necrosis. For the purposes of accurate diagnosis, systemic signs such as characteristic of viper bites, while signs of neurotoxicity and muscle damage (rhabdomyolysis) are primarily common with elapid bites and sea snake bites, respectively. There may, however, be some exceptions to these observations pertaining to snakebites. For instance, bites from some viper snakes, such as the tropical rattlesnake and berg adder, exhibit signs of neurotoxicity in patients with systemic envenomation. Following the bite of vipers particularly, localized effects including necrosis may occur; however, necrosis tends to manifest slightly later on and therefore may not necessarily be of diagnostic value. Also, in addition to neurotoxicity, certain Australian elapids can cause incoagulable blood and hemorrhage.  Whether or not fang marks are present cannot be used as a basis for diagnosis albeit the distance between the fang marks may offer a clue about the size of the snake implicated in the bite. Nonetheless, the identification of fang marks does not form sufficient basis to suggest that venom has indeed been introduced because in close to 50% of bites, no venom is injected [14].

#### *3.3.2 Laboratory and other diagnostic methods*

The diagnosis of snakebites in the laboratory operates on the basis of the changes, which take place following envenomation in victims. Such changes may include detecting irregular changes in blood parameters [such as incoagulable blood assessable using the 20-minute whole blood clotting test (WBCT20), dramatic drop in platelet count, and changes in white and red blood cell counts], changes in some enzyme levels (e.g. creatine phosphokinase), and presence or absence of myoglobinuria, as well as detecting specific venom antigens in the blood of envenomed individuals (biodetection techniques using immunologically based methods) [14].

Severe hemorrhage may be indicated by a low hematocrit value, while a high hematocrit value may be indicative of haemoconcentration, which occurs when plasma leaks into the tissues due to an increase in capillary permeability. Systemic envenoming by viperids, non-front-fanged colubrids, and oceanic elapids are mainly characterized by incoagulable blood. The 20-minute whole-blood clotting test (WBCT20) is a simple test that involves collecting a few milliliters (normally 2 mL) of venous blood into a new, clean, and dry glass tube, allowing it to stand undisturbed for 20 minutes at room temperature and then tilting the tube to see if it has clotted [25]. The absence of clotting is an indication of a venom that is anticoagulant or severe consumption coagulopathy [23]. Generalized fibrinolysis and intravascular coagulation may be detected by more sensitive laboratory tests including prothrombin and


**Abbreviations**

ABTS—2,20 -azinobis (3-ethylbenzothiazoline-6-sulfonic acid) ASE—accelerated solvent extraction ChCl—choline chloride DES—deep eutectic solvent DM—dry material DPPH—2,2-diphenyl-1-picrylhydrazyl radical FRAP—ferric reducing antioxidant power GAE—gallic acid equivalents HBA—hydrogen bond acceptor HBD—hydrogen bond donor LacA—lactic acid MAE—microwave-assisted extraction NADES—natural deep eutectic solvent PLE—pressurized liquid extraction SFE—supercritical fluid extraction TEs—Trolox equivalent UAE—ultrasound-assisted extraction