The implementation of participatory democracy in French communes

Keywords: participatory democracy, socialization, territorial configuration, size of the commune, local consultations

the purpose of the article is to analyze how participatory democracy has been implemented in French communes. The question of neighbourhood democracy is a norm of political speech as local representatives need to legitimize their role in public space by getting closer to their electors. Does the implementation of participatory tools depend on the ideology and the socialization of the mayors or on the structure of the commune (size, territorial configuration)?

The paper will mainly use a quantitative investigation made on communes above 5 000 inhabitants as well as interviews conducted in specific communes which were known for their participatory culture.

direct democracy includes all the mechanisms which interfere with the decisions (referenda, popular initiatives, face-to-face relations) and participatory democracy highlights the deliberative aspect (Fishkin, 1995: 4). In other words, participatory democracy is the name given to all political systems between pure direct democracy (citizens take directly part in main political decisions) and pure representative system (citizens just choose people in charge of public duties). These two extremes work as ideal-types because participatory elements and representative system are mixed in the reality (Budge, 1996: 44). In fact, participatory democracy is composed of a multitude of tools which affect the political process at different stages. For instance, the consensus conferences aim at selecting a panel of citizens who become well-informed of scientific problems [START_REF] Boy | Un exemple de démocratie participative : la "conférence de citoyens" sur les organismes génétiquement modifiés[END_REF] whereas the district councils focused on the micro-level deliberative aspect. All tools cannot be simply compared as the question of scale is crucial: who is eligible to take part in the process (Weale, 1999: 146)? What for? When and how long? (Falise, 2003: 52-53).

Participatory democracy involves specific citizens who help politicians to take a decision on a particular topic (Bourg, Boy 2005: 5).

The topic of participatory democracy has been more and more popular among citizens and representatives in France who regard it as a way of strengthening and legitimizing political decisions thanks to organized discussions. The local level seems to be appropriate to the experimentation of participatory processes. We would like to analyze how participatory democracy has been implemented at local level that means how French communes created specific participatory instruments which involve inhabitants in local affairs. First, we will use macro-level aggregate data on political attitudes of French citizens thanks to the questionnaires of World Values Survey. How do French citizens perceive their relation to political participation? We will compare the attitude of citizens with the series of laws concerning participatory democracy from the nineties on: did the representatives try to adapt to the shift of values of the citizens? Second, we will use a database made on participatory tools in French communes above 5 000 inhabitants in 2006 in order to appreciate the influence of political context and the territorial configuration in the implementation of participatory tools. Is there a correlation between the size of the commune, the territorial configuration, the socialization of local representatives and the implementation of participatory democracy? These data will be completed by half-directive interviews made with local representatives of communes that developed a specific participatory culture.

The research design is inspired by the configurative comparative approach developed by Charles C. Ragin [START_REF] Ragin | La place de la comparaison: jalons pour la recherche comparative configurationnelle[END_REF]) which combines quantitative and qualitative studies. The quantitative overview is important in order to define a general trend and find out specific cases. We extracted the cases of communes which developed a participatory culture by promoting a panel of instruments and proceeded different interviews with political actors involved in participatory democracy. The final goal is to study the way local politicians implement participatory instruments [START_REF] Unger | Democracy realized, the progressive alternative[END_REF].

Political attitudes and values in France

The political attitude of citizens can be measured through specific data extracted from World Values Survey. We are interested in analyzing non-conventional participation that means specific mobilizations outside traditional organizations such as political parties or trade-unions. Conventional participation can be expressed through different political acts such as signing up petitions, taking part in demonstrations and boycotts, occupying factories' buildings, organizing strikes. These specific acts reveal citizen mistrust towards political institutions: this is why we deal with values of participation when we consider this specific repertoire. Pippa Norris noticed that the non-conventional participation has become a norm in most of the western societies (Norris, 1999: 263). Based on an analysis of World Values Surveys (WVS) data in 44 countries, she noticed that the more citizens got involved in political matters, the less they claimed any political belonging and the less they had trust in institutions. In the data concerning France in the fourth wave of WVS 1 , 34% of the answerers declared that they took part in a peaceful demonstrations, 13% said that they had boycotted a product in the past; 11.2% took part in an unofficial strike and 8.3% organized a sit-in in a factory to support a protest. These answers show that the non-conventional participation has been more and more important in France. If we compare these data with other countries such as Germany 2 , then we see the difference in the modalities of participation as only 2% of them took part in unofficial strikes and 1.3% occupied factories´ buildings. The repertoire of protest is stronger in France as there is a prevailing mistrust in political institutions.

The importance of non-conventional participation (Taniguchi, 2006: 416) illustrates the shift of values among French citizens. In the WVS, a special indicator was created to study the post-materialistic trend (care for environment and quality of life, mistrust in political authority, promotion of other ways of engagement in politics). The emphasis on new ways of taking part in politics is a characteristic of post-materialistic values. When we look at the results presented in table 1, we observe that around 20% of the respondents share postmaterialistic values and 27% of them materialistic values. The majority of them shares mixed values, which means that materialistic and post-materialistic attitudes coexist and vary with the topic [START_REF] Inglehart | Modernization, cultural change, and the persistence of traditional values[END_REF]. Some respondents wish to have more participation in the political process while they support a clear authority in the political system. Those attitudes can reflect conflict of values: as a matter of fact, they show that the shift of values is an ongoing process in France and that the citizens are critical about the political institutions. The DATA FILE, 1981-2004, v. 20060423, 2006. Aggregate File Producers: Análisis Sociológicos Económicos y Políticos (ASEP) and JD Systems (JDS), Madrid, Espagne/Tilburg University, Tilburg, Pays-Bas. Data Files Suppliers: ASEP and JDS, Zentralarchiv für Empirische Sozialforschung (ZA), Cologne. Aggregate File Distributors: ASEP, JDS and ZA. The total number of respondents is 3680 for France. 2 The number of answerers is 7268 for Germany.

French local elites are aware of this evolution and adapted themselves to it by allowing citizens to take part in the elaboration of public policies. This is how we can understand the institutionalization of participatory tools since 1992 in France.

The context of participatory democracy A series of laws has been voted in France since 1992. A law enacted on 6 February 1992, authorised the consultation of "inhabitants" for a limited number of topics. Many extramunicipal commissions were introduced and the participation of inhabitants was recommended. The concept of consultative popular initiative was introduced for the first time on 4 February 1995. One fifth of the electors can request the organization of a local consultation. District councils (conseils de quartier) were created and the law of 27 February 2002 made them compulsory for cities having more than 80 000 inhabitants. Recently, the second Act of Decentralization of 1 August 2003 recognized the binding local referendum and dealt with the idea of citizen participation. The table 2 reports the use of the expression "participatory democracy" in the parliamentarian debates since 1986. We observe that the topic was deeply discussed between 2001 and 2003, especially because of the law of 27 February 2002 and the second Act of Decentralization. The topic is nowadays prevailing in local politics and this is why we focused on the degree of the implementation of participatory democracy in French communes. Participatory democracy makes politics at local level more complex as mayors have to adjust their agenda to the problems of inhabitants. There is a "'built-in-tension'" between output-and input-orientation: intentions to enhance local government efficiency might create problems for democratic inclusiveness and vice-versa" (Steyvers, Pilet, Reynaert, Delwit, 2005: 24). Participatory tools have been created in France to make citizens take part in the definition of adequate local policies.

Between 1992 and 2003, different participatory tools have been created such as local consultative council of wise men (conseils des aînés), children local councils (conseils municipaux d´enfants), young councils (conseils de jeunes), district councils (conseils de quartier), councils of foreign communities (conseils des communautés étrangères), urban planning sessions (ateliers d´habitants) and diverse consultative councils. There are a lot of different labels for participatory tools as the association for social and local democracy (ADELS) counted down 184 different names for participatory instruments 3 . In a survey made on 140 directors of communication of French communes above 10 000 inhabitants, a question referred to the implementation of participatory democracy: 26% of them declared that they had young councils in their communes, 20% councils of wise men and 5% councils of foreign communities4 . Political elites use participatory democracy as a means to create a feeling of local community which can be useful to legitimize their power and renew their image among electors. The quantitative analysis on participatory tools gives an accurate idea of the new participatory policies promoted by French mayors.

An empirical study

The present study focuses on the implementation of participatory democracy in 2006 in the 1881 communes which have more than 5 000 inhabitants. In order to build the sample, we referred to the website of Profession politique, which is an exhaustive website edited by the Société d'édition publique (SEP), a press group specialized in information about public life. Profession politique is a weekly magazine5 and presents a database on French and European political life. All the data on the French political personnel are regularly updated.

This database includes different sociological and political information on mayors from municipalities with more than 2 500 inhabitants and is based on the census made by the INSEE in 2004. The table 3 deals with the size of the French communes above 5 000 inhabitants: around 87% of the communes of the sample have between 5 000 and 30 000 inhabitants. The table 4 gives the political belonging of the mayors of all these communes: 19% of mayors do not claim any political belonging, they won their mandate without referring to any political party; 29% of the communes between 5 000 and 10 000 inhabitants have such mayors. The bigger the commune is, the more the mayors are engaged politically.

We added to the main conservative party (UMP) the mayors from the right wing who are not members of any political party and to the Socialist party the mayors from the left wing without any political membership. We observe in 2006 a balance between the right wing and the left wing (stability of the cleavage right / left) for the entire sample.

The territorial configuration

The territorial configuration (Elias, 1991: 154-61) includes the size of the city and its institutional environment (population density, inter-municipal structure). In the field of political science, Robert A. Dahl initiated a work on the topic of size and democracy in the seventies [START_REF] Dahl | Size and Democracy[END_REF]. These investigations are still pursued with the works of Larsen [START_REF] Larsen | Municipal Size and Democracy: A Critical Analysis of the Argument of Proximity based on the Case of Denmark[END_REF] and Le [START_REF] Roy | Participation, Size, and Democracy: Bridging the Gap between Citizens and the Swedish State[END_REF]. Robert A. Dahl points out the difficult relation between participation and the size of units: "for most citizens, participation in very large units becomes minimal and in very small units it becomes trivial. At the extremes, citizens may participate in a vast range of complex and crucial decisions by the single act of casting a ballot; or else they have almost unlimited opportunities to participate in decisions over matters of no importance. At the one extreme, then, the people vote but they do not rule; at the other, they rule -but they have nothing to rule over" (Dahl, 1971: 97). He establishes an explicative model that we can sum up in an antagonistic relation between the control of political decisions by the citizens and the efficiency of the public policies. Those two sides vary with the size of political systems. The higher the size increases, the higher the efficiency is but the poorer the citizen participation gets. This size effect is not significant because of the size of the country but because of the distribution of units inside the country (Dahl, Tufte, 1974: 65). If both authors included in their study other factors as political culture, we mainly tested empirically the influence of size by applying it to the French municipalities of more than 5 000 inhabitants.

There might be a relation between the size of the commune, the identity of the commune in its environment and the implementation of participatory democracy. The size of the commune and the territory configuration are the two main keys studied in order to observe how this implementation is made. In his work on local system, Albert Mabileau proposed a typology of four models of cities [START_REF] Mabileau | Le système local en France[END_REF] that we took in order to elaborate our indicator of urban stratification. In France, 90% of the communes have less than 2 000 inhabitants; as a matter of fact, most of the communes are rural, but this criterion does not take the question of configuration into account. Two communes of the same size can have opposite configurations. This is why the concept of territorial configuration is the most appropriate tool in our investigation.

The mayoral leader is seen in interdependency with his communal environment which is composed of his own political and administrative staff, the communal society and the institutional environment (the type of intercommunalité and the relation of the commune to this structure). In other words, the mayors might choose to implement participatory democracy tools in order to strengthen their leadership on the local territory. We also assume that there is a certain stratum of low-sized and middle-sized cities which specifically implement participatory democracy tools. Those political leaders are more willing than the others to implement them. In some metropolitan areas, the implementation of participatory tools will not cause the transformation of political order (Jouve, 2005: 336) as it could do in smaller cities. In the seventies, Robert A. Dahl and Edward R. Tufte refused to find out an optimum size for political participation. "Can we say, then, that there is any optimum size for a political system? In our view, the answer is clearly no. Different problems require political units of different size" (Dahl, Tufte, 1974: 135). This negative statement answers to the second of their three main questions in their research: "more specifically, is 'democracy' related in any way to size? How large should a political system be in order to facilitate rational control by its citizens? What are the comparative advantages and disadvantages enjoyed by political systems of different sizes?" (Dahl, Tufte, 1974: 1). They established a link between the size and the two elements of a democratic system, on the one hand the citizen participation and the feeling of influencing the political process (sense effectiveness) and on the other hand the capacity of the system to answer efficiently to the social demand and solve the complex problems of society. If participation is trivial in small urban units, the big units are more efficient to emit the adequate public policies even though the citizens do not have so many occasions to interfere in the decisions outside the vote. In the current research, the goal is to know whether there is an optimal size of cities which implement participatory democracy.

The socialization of mayors

The second hypothesis points out the socialization of mayors. There might be a relation between the date of entrance in local politics and the implementation of participatory democracy. The new generation coming from the associative and political world (Mabileau, 2002: 142) might be more willing to use this resource. The local notables, who are deeply rooted in the political system and who have an ancient experience of the mechanism of local politics, are socialized in an elitist political culture which refuses any delegation of power.

The cumul des mandats could reinforce a conception of representative democracy. This is why those mayors would not implement those kinds of tools (Marrel, 2004: 122-28). The more functions the mayor cumulates, the more monopole he has and the less predisposed to implement participatory instruments. This assumption needs to be tested as we may say the contrary: the more a mayor cumulates, the more he needs to justify his mandate in front of his electors and the more he might implement participatory tools. The mayors who cumulate can use participatory tools as a way of legitimizing their local power while they have different political tasks.

Data and variables

To test the hypotheses, we used two sets of indicators, the ones related to the dependent variable (the implementation of participatory tools) and the others that make up the independent variables (socio-political indicators about mayors and the structure of the commune).

Dependent variable: the implementation of participatory democracy

There are many participatory tools that French mayors implemented. We collected the data from the presentation of these tools through the website of every commune above 5 000 inhabitants. This variable is ordinal and equals 0 if there is no participatory tool, 1 for one participatory tool and n for n instruments. In order to strengthen the indicator on the participatory democracy, we added an associated indicator which is the local consultation. In France, the local consultation was introduced in the law of 6 February 1992 before the law of 1 August 2003 recognized the decisive local referendum when 50% of the local population takes part in the vote. There have been 102 communes which have been holding at least a consultation since 1992. Almost all the local votes outside elections are consultative: the mayors, who control this instrument, want to make people deliberate on a question. At the same time, they increase their legitimacy (Paoletti, 1997: 234-235). This is why we took local consultations as another way of promoting a participatory strategy. The question is to see whether in those communes, the local consultation was used for itself or in connection with other participatory instruments.

Independent variables

In order to make up the indicator on the size of communes, we distinguished six levels (see table 3). We started from the typology made by Albert Mabileau with the four levels, the big cities, the métropoles d'équilibre, the middle-sized cities and the small cities. In order to be exhaustive, we added two scales to this typology. Then, we built some indicators on the territorial configuration to study their effects on the implementation on participatory democracy. The first one is about rural life, which is defined by three traditional axes according to Jean-Claude Bontron: economical (agricultural activities), sociological (values based on family and the attachment to the farm) and demographical (weak density of population) [START_REF] Bontron | L'irrésistible élan de l'intercommunalité[END_REF]. The decline of rural societies is explained by the diffusion of an urban life standard and the decrease of rural families in the countryside: there is only the demographic factor which can be taken into account for the difference between urban and rural areas. As a matter of fact, according to the figures from the census of INSEE (French national independent statistics) made in 1999, we created a variable which measures the density of the district where the commune is located. In order to test the impact of the intermunicipal structures, we established an ordinal variable from the smallest structure to the biggest one: for the communes without any cooperation, the variable scores 0, then 1 is for the integration in a communauté de communes, 2 in a communauté d'agglomération, 3 in a communauté urbaine or a syndicat d'agglomération nouvelle: 247 communes are not integrated in an inter-municipal structure (13.2%), 687 communes belong to a communauté de communes (36.5%), 754 to a communauté d´agglomération (40.1%) and 192 to the third category (10.2%).

For the indicators testing the impact of socialization of the mayors on the implementation of participatory democracy, we included the age with the birth date of the mayors, the date of their first mandate as mayors and then the first time they had a responsibility in the field of local politics (member of local council, regional counsellor, member of the executive of the department). To test the relation between the cumul des mandats and the implementation of participatory democracy, we created a variable whose goal is only to measure the vertical cumul (other mandates) and not the horizontal ones (other professions). We only took elective mandates and positions of president or vice-president of inter-municipal structures into account as these last functions have been really important (Marrel, 2004: 128) in the reorganization of cumulative strategies since the law on cumul des mandats of 2000. It is not the type of mandate which is important but the cumulative strategy.

Empirical results

The hypotheses are tested through a linear regression with robust standard errors presented in table 5. First, there is a correlation between the size and the implementation of participatory democracy. With a significant Beta coefficient of 0.29, it is even the most explicative factor. Indeed, because we excluded rural communes from our sample, we should have found a negative correlation between the implementation of participatory democracy and the size of the communes. This is due to the resources of the communes: the correlation indicates that there is a systematic use of participatory instruments when the commune has more resources, even if the commune is not compelled to develop the participatory instruments. This model shows the diffusion of the participatory norm on the whole territory which is mainly used as a way of legitimizing the work of the representatives. The bigger the commune is, the more it focuses on participatory democracy as a way of getting closer to the electors.

The increasing tendency to implement participatory elements shows that the link between the size and democracy is not negatively correlated. The correlation confirms that there is no specific size where the efficiency of democracy is balanced by its inclusiveness. If the bigger cities do have the best system capacity, the low-sized cities do not have the monopole of citizen efficiency, as Robert A. Dahl and Edward R. Tufte perceived it thirty years ago: "no single type or size of unit is optimal for achieving the twin goals of citizen effectiveness and system capacity. For in the present world, and even more surely in the future, democratic goals conflict, and no single unit or kind of unit can best serve these goals" (Dahl, Tufte, 1974: 138). The other explanation comes from the structure of big cities: local democracy exists in differents areas such as the arrondissements in Paris. We conducted interviews in different city halls of arrondissements (mairies d´arrondissement) on participatory tools which were implemented. Local democracy is more complex and the units can be fragmented when the commune is quite big. In Paris, local democracy was all the more possible in these arrondissements as the political affiliation of the mayor and certain mayors of arrondissements was not the same 6 . As a matter of fact, participatory democracy is a resource which is at stake in the electoral competition.

The second important explicative factor is the type of inter-municipal structure. The beta coefficient (0.09) is not as high as the one of the size but is above all the other variables. This is to be explained by the fact that the commune does not loose its identity inside a bigger complex. The participatory democracy is then a major resource to build this identity. This result is reinforced by the proximity with Paris: when a commune belongs to the suburb of Paris, it tends to implement participatory democracy in order to strengthen local life and the relations between citizens and representatives. We could verify this fact by conducting specific interviews in communes near Paris which were engaged in the implementation of 6 Interview conducted with Guillaume Métayer, administrative director of the 19 th arrondissement (4 June 2004) and Cyril Egner, green party member and vice-mayor of the 3 rd arrondissement (5 March 2004).

participatory democracy such as Arcueil, a commune of around 20 000 inhabitants in the south part of Paris. The vice-mayor of Arcueil explained us in an interview how participatory democracy was implemented in the commune: "there are thousand ways of taking part in local life in Arcueil. The district councils (conseils de quartier) evolved, we have more and more autonomy; the inhabitants begin to administrate themselves these councils and their budget. Since 1995 and 1996, some members of the local council (conseil municipal) have been appointed to these district councils in order to transmit the summary of these citizen assemblies to the local council. The agenda of these district councils is prepared both by the selected representatives (élus référents) and the citizens […]. As far as the local council is concerned, we propose different ways of associating inhabitants to the political decisions which affect their environment. […] In 1997, we created the council budgets; the inhabitants decide how they use this money"7 . In this interview, we perceived that the participatory democracy was seen as a top-down process which aimed at stimulating the citizen commitment. It implies the definition of specific instruments and rules in order to be efficient. Participatory democracy is defined as a method of local governance8 : the closer the representatives get to the problems of citizens, the more efficient they are in the selection of appropriate local policies.

The sociological variables have a weaker coefficient of significance than the variables of territorial configuration and size. Their explicative role is thus secondary but still gives some important information. There is no correlation between the cumul of mandates and the implementation of participatory democracy: 72% of the mayors who do not practise the cumul implemented participatory tools against 77% of the ones who have several mandates. The mayors tend to use participatory tools as a strategy which corrects the image of distant politicians. On a survey conducted on European mayors of communes above 10 000 inhabitants, 188 French mayors were interviewed (20 % of the French communes above 10 000 inhabitants): 27 mayors have been in office in more than five years. On the whole, they approve participatory tools such as neighbourhood panels (76.3%)9 . The political experience is not necessarily opposed to the implementation of participatory democracy.

When we look at the relation between the first time the mayors were elected and the implementation of participatory democracy, we notice that the election of 1977 was the first step in this direction: 77% of the mayors elected in 1977 and who still govern, introduced participatory tools [START_REF] Wargny | Mairies frappées d'autogestion[END_REF], 71% of the ones elected in 1983 and 70% of the ones elected the first time in 1989. The 1977 election was characterized by the entry of leftist mayors who dealt with the ideas of participatory democracy; 77% of the mayors elected in 2001 implemented those tools. The duration of the mandate (cumul in time) is not necessarily contradictory to the implementation of participatory tools. On the contrary, mayors who wish to be elected again, create participatory tools to renew their image. The age of the mayors does not have any influence on the implementation of participatory democracy. In fact, the question of political generation is more important in the implementation of participatory democracy. The mayors who came to power before 1977 are less interested in the promotion of these tools.

The political cleavage does not have any impact on the implementation of participatory democracy as it has become a norm shared by different political parties with a variable intensity. The table 6 shows that all the main political trends are affected in the same proportion by the participatory norm. Participatory democracy is not a discriminating topic at local level in France whereas the left-wing parties are more willing to promote it at national level. The difference is in the depth of the participatory reforms conducted at local level: some communes implemented several tools which complete each other. In Arcueil, the preparation of the local consultation of 5 December 1999 on the future of the city was possible because of the connection between different participatory tools: "we decided to hold a referendum on an important issue in local life which is the restructuration of the centre of the city. To prepare it, we organized different debates and we used the district councils as deliberative arenas on the way we should restructure the centre"10 . The participatory strategy consists in combining tools so that the citizens can take part in the local political debate on the future of the city.

The configurative comparative approach was used in order to investigate the communes that developed several participatory tools and held local referenda. In these communes, the participatory strategy depended on the will and personality of the mayor like Mons-en-Baroeul in the suburb of Lille (29 000 inhabitants). This commune is the most participatory one in France: four local referenda have been held since 1977, a council of foreigners was created in 1987 as well as other participatory tools. Marc Wolf was the mayor of the commune between 1977 and 2001; he was a member of the Socialist Party and believed in the ideas of self-government. "The left-wing was aware of the importance of the municipal action not only to prepare its access to the government but in itself, in the making of a socialist project based on self-organization" (Wolf, Osselin, 1979: 10). Marc Wolf devoted himself to the implementation of participatory democracy: he decided to hold referenda on local taxes (these referenda were not legal) and let citizens choose the type of local policies that they wanted to have. As the former vice-mayor of Marc Wolf told us, "the referendum system is not easy and not mooted in the French political system. Some citizens prefer to let representatives act instead of making choices. The personality of the former mayor was contested: some people reproached him to be too rigid, but he had an authentic political engagement […]. As far as I am concerned, Marc Wolf embodies another way of doing politics"11 . The implementation of participatory tool is linked to the personality of the mayor and its participatory ideology, it is not a bottom-up process.

The organization of local consultations is another way of implementing participatory democracy but unlike the other tools, local consultations are not regular. We notice that 102 mayors held a local referendum between 1992 and 2006 (six communes have held two consultations since 1992 and two communes have known four consultations). Most of them are not binding even though the law opens both possibilities (Premat, 2005: 204). Out of these 102 communes, 88 had implemented participatory tools which underlines the connection between local consultations and participatory tools. Local consultation is seen as one of the instruments which can enlarge the participatory repertoire. According to the survey conducted on European mayors, 80% of French mayors perceived the local referendum as an appropriate tool of communication between the local leaders and the population12 . In reality, they used it almost in a consultative way: there have been no binding referenda held in French communes above 10 000 inhabitants since the Law of 1 August 2003. In one of the interviews made in Arcueil, a vice-mayor deplored the ambiguous notion of citizen involvement: "We have to go further in the implementation of participatory democracy. We have to associate citizens to the local decisions, it is an openness to the political decision.

Only the representatives decide, we do not mix types of governance. We have to make citizens understand that they can influence the decisions, but they do not decide instead of the politicians. We are not enough accurate when we talk to people. We let them believe that democracy is participatory, but we are not sure that the tool will create participation. We can be wrong if there is no participation [...]. The participatory democracy is under the control of the representative system"13 . This is why the implementation of participatory tools is not really deep in France: only consultative tools are promoted in order to control the citizen participation. French local representatives share the republican ideology and do not believe in the real empowerment of citizens [START_REF] Genieys | The Sociology of Political Elites in France: The End of an Exception?[END_REF].

In a nutshell, the territorial configuration and the size highlight the communicative policies of mayoral leaders in terms of participatory democracy much more than the political career of those leaders. The question of scales and the location of the commune are determining variables in the implementation of participatory democracy which legitimizes the work and the visibility of local leaders. Paradoxically, participatory democracy is not a property of low-sized communes. The big cities use resources in order to implement participatory democracy. Surprisingly, the cumul does not have any influence on the use of participatory implementation. We thought that some local leaders who would have different mandates could have created participatory tools in order to compensate their busy schedule and difficult availability. The mayors also tend to implement participatory policies when their commune belongs to a big inter-municipal structure. There is a relation between territorial configuration and democracy and not only between size and democracy. The administrative environment of the commune has an influence in the implementation of participatory democracy.

Last but not least, the local consultation is a convenient participatory tool as the mayors control it entirely: at the end of 2005 no decisional referendum was organized (Koebel, 2006: 101). Local consultations are like surveys that mayors use to know the opinion of the citizens. They are mainly used by mayors who implemented participatory tools. On the whole, French local representatives adapted to the participatory norm by implementing tools which are merely consultative. They control the participatory process which shows that participatory democracy reveals the transformation of the representative system: the politicians are more responsive but do not share the main decisions with citizens.

The other point is that participatory democracy is a gadget for the mayors who legitimize their local power. The fragmentation of local powers (36 000 communes, 26 regions, 100 départements) is a weakness of the French administrative system. The inter-municipal bodies are a response to the effectiveness of local policies but they are not visible for the citizens (Kerrouche, 2008: 124). The surveys made on citizen values revealed that the citizens were more and more critical towards political institutions; the implementation of consultative tools might not satisfy their wishes to take part in the selection of adequate of public policies. *Our acknowledgments go to Niall McCourt for his review of the article. DATA FILE, 1981-2004, v. 20060423, 2006. Aggregate File Producers: Análisis Sociológicos Económicos y Políticos (ASEP) and JD Systems (JDS), Madrid, Espagne/Tilburg University, Tilburg, Pays-Bas. Data Files Suppliers: ASEP and JDS, Zentralarchiv für Empirische Sozialforschung (ZA), Cologne. Aggregate File Distributors: ASEP, JDS and ZA.

1

  European Values Study Group and World Values Survey Association. EUROPEAN AND WORLD VALUES SURVEYS FOUR-WAVE INTEGRATED

Table 1 :

 1 Principal values of French citizens World Values Survey, Fourth vague 14 14 European Values Study Group and World Values Survey Association. EUROPEAN AND WORLD VALUES SURVEYS FOUR-WAVE INTEGRATED

	Index(*) Pays France	Materialistic values 1 000 Source:	Mixed values 1 937	Post-Materialistic values 743	Total 3 680

Table 2 :

 2 Apparition of the expression "participatory democracy" in French parliamentarian debates since 1986 of the debates thanks to the websites of the Senate and the French Assembly

		Senate		National Assembly
	Year 2006 2005 2004 2003 2002 2001 2000 1995-1999 1986-1994 Total 1986-2006	Bills and Laws 24 25 10 13 6 4 0 2 0 84 Source: analysis	Laws' resolutions' proposals, proposals, convents, questions 4 2 0 0 2 1 1 0 0 10	Parliamentary discussions 0 2 7 117 52 127 0 11 0 316

Table 3 :

 3 The size of the French communes above 5 000 inhabitants

	Size of the commune Between 5 000 and 10 000 inhabitants Between 10 000 and 30 000 Between 30 000 and 50 000 Between 50 000 and 100 000 Between 100 000 and 200 000 Above 200 000 Source: Database 2006, IEP Bordeaux (in cooperation with Julien Dewoghélaëre) Number of communes Proportion 960 51% 666 36% 137 7% 78 4% 29 1.5% 11 0.5%

Table 4 :

 4 The political belonging of mayors from French communes above 5 000 inhabitants in 2006

	Political affiliation Without claiming any political belonging UMP (conservative party) UDF (Liberal Party) Far-right Party (FN) Communist Party Socialist Party Green Party Local Parties Total Source: Database 2006, IEP Bordeaux (in cooperation with Julien Dewoghélaëre) Number of mayors Proportion 355 19% 615 32.8% 116 6.2% 2 0.1% 145 7.8% 615 32.8% 11 0.6% 14 0.7% 1873 100%

Table 5 :

 5 The explaining factors of the implementation of participatory democracy in French communes

	Number of observations = 1491

Source: Database 2006, IEP Bordeaux (in cooperation with Julien Dewoghélaëre)

Table 6 :

 6 Political affiliation and implementation of participatory tools in French communes of more than 5 000 inhabitants

	Degree of implementation Without any political affiliation One participatory tool 199 Two participatory tools 44 Three participatory tools 7 Four participatory tools 7 Proportion of mayors 72.4% implementing participatory tools Source: Database 2006, IEP Bordeaux (in cooperation with Julien Dewoghélaëre) UMP PS PCF UDF 352 331 66 62 84 86 17 19 28 38 8 5 11 22 12 4 77% 77.6% 71% 71.4%

Territoires, nº441, October 2003.

Survey made by the firm Opinionway between 1 and 13 November on 140 directors of communications of communes above 10 000 inhabitants. The survey was made by CAWI System (Computer Assisted Web Interview). There are 921 communes above 10 000 inhabitants in France.

http://www.professionpolitique.info

Interview conducted with Anne Buyck, vice-mayor ofArcueil (16 December 2003).

In an interview conducted in the commune of Lanester (Bretagne, 23 000 inhabitants), the director of communication dealt with the "participatory philosophy" that the mayor developed for the commune. Interview conducted in Lanester on 13 September 2006.

Database SSD 0822 « The European mayor -Political leaders in the changing context of local democracy », Henry Bäck, School of Public Administration, Göteborg University.

Interview of the vice-mayor of Arcueil, Anne-Marie Buyck, 16 December 2003.

Interview made in Mons-en-Baroeul on 3 December 2005.

Database SSD 0822 « The European mayor -Political leaders in the changing context of local democracy », Henry Bäck, School of Public Administration, Göteborg University.

Interview conducted with Denis Weisser on

December 2003, socialist vice-mayor of Arcueil.


Mark Behzad Doost
email: doostmb@gmail.com

Foldy-Wouthuysen WKB transfer matrix method for Dirac tunneling through monolayer graphene with a mass gap

Keywords:

I provide a transfer matrix method for the Foldy-Wouthuysen representation of the Dirac equation. I develop the relationship between the reflection and transmission coefficients of the Dirac spinors and the wavefunction in the transformed representation. I develop a WKB approximation for Dirac fermions which has the same elegant form as the WKB solution to Schrödinger's equation. My WKB approximation is to all orders and includes the semi-classical turning point. I provide an extension to fully 2-dimensional periodic structures by Fourier methods for band engineering. I verify my methods for all energies by comparison with analytic solutions developed in the Dirac spinor representation. I also include a rich appendix detailing my research into the Green's functions of Dirac fermions.

Introduction

In Refs. [START_REF] Barbier | Extra dirac points in the energy spectrum for superlattices on single-layer graphene[END_REF][START_REF] Barbier | Dirac and kleingordon particles in one-dimensional periodic potentials[END_REF][START_REF] Arovas | Dirac spectrum in piecewise constant one-dimensional (1d) potentials[END_REF] the pseudo-relativistic dispersion of 2-dimensional (2D) materials was investigated theoretically for the emergence of extra Dirac points with 1D periodic barriers. The method employed was the application of Bloch's theorem to transfer matrices of Dirac spinors. These band engineering studies attracted considerable interest since the vanishing band gap of 2D materials is an Achilles heel for applications. Other methods for band engineering that open the way to creating 2D semiconductor materials include strain engineering [START_REF] Guinea | Energy gaps and a zero-field quantum hall effect in graphene by strain engineering[END_REF], vertical stacking [START_REF] Novoselov | 2d materials and van der waals heterostructures[END_REF], and chemical doping [START_REF] Wan | Flexible n-type thermoelectric materials by organic intercalation of layered transition metal dichalcogenide tis 2[END_REF].

Armed with the knowledge to create semiconductor 2D materials a theoretical study of the operational window for negative differential resistance was pursued by other authors [START_REF] Song | Negative differential resistances in graphene double barrier resonant tunneling diodes[END_REF]. The mathematical method employed in their study was the application of transfer matrices to the Dirac spinors of a rotated Dirac Hamiltonian derived in Ref. [START_REF] Sonin | Effect of klein tunneling on conductance and shot noise in ballistic graphene[END_REF].

Motivated by the study of unconventional transmission and density of states, the modeling of Dirac fermions has moved beyond unrealistic square barrier systems. In experimental devices barriers are more often smoothly varying, semi-classical (WKB) methods are needed. An approach using separate Hamiltonians for electrons and holes in Ref. [START_REF] Tudorovskiy | Chiral tunneling in single-layer and bilayer graphene[END_REF] gave way in Ref. [START_REF] Zalipaev | Resonant tunneling and localized states in a graphene monolayer with a mass gap[END_REF] to an expansion in powers of ℏ partly resembling the WKB method for the Schrödinger equation. However it is acknowledged by the authors of Refs. [START_REF] Tudorovskiy | Chiral tunneling in single-layer and bilayer graphene[END_REF][START_REF] Zalipaev | Resonant tunneling and localized states in a graphene monolayer with a mass gap[END_REF] that their mathematical approaches to semi-classical analysis can be divergent at the classical turning point.

The contribution to the theory of tunneling in 2D materials that I present in this article is the application of the Foldy-Wouthuysen (FW) representation of the Dirac equation.

Whilst the FW transformation was first conceived as a semi-relativistic representation of the Dirac equation [START_REF] Foldy | On the dirac theory of spin 1/2 particles and its non-relativistic limit[END_REF], I will show, through my rigorous derivations in Section 2 to Section 13 and my analytic and numerical verification of Section 14 and Appendix B, that the FW representation remains an exactly accurate representation of the Dirac equation for all energies.

I find that the FW representation allows a derivation of the relativistic WKB approximation, Section 7 to Section 10, in the same elegant form as the approximate WKB solution to Schrödinger's equation. Since at low energies the FW representation reduces to the Schrödinger equation, I have found, in Section 11, a simple method to mathematically describe the connections between WKB regions.

Fourier analysis is used in Section 12 to apply my approach to fully 2D periodic structures. Section 12 offers the prospect of extending the scope of the band engineering studies made in Refs. [START_REF] Barbier | Extra dirac points in the energy spectrum for superlattices on single-layer graphene[END_REF][START_REF] Barbier | Dirac and kleingordon particles in one-dimensional periodic potentials[END_REF][START_REF] Arovas | Dirac spectrum in piecewise constant one-dimensional (1d) potentials[END_REF]. My numerical and analytic results showing that the FW equation is an exactly accurate representation of relativistic fermions, for all energies, will be important here. The Fourier transform method in Section 12 requires expansion of the wavefunction in plane waves of momentum beyond semi-relativistic energies.

In Appendix E I make the reader aware of a limit to appealing for solutions of the Dirac equation from its FW representation, I reveal and discuss a Green's function (GF) paradox arising when transforming between the two representations.

My article is organized as follows: Section 2 outlines the derivation of the FW representation from the Dirac equation. Section 3 gives the transmission of the Dirac equation in terms of the transmission of its FW representation. Section 4 derives the boundary conditions of the FW equation at sharp steps in potential. Section 5 and Section 6 develop the transfer matrices for square barriers and delta potentials. Section 7, Section 8, Section 9 and Section 10 give a 1𝑠𝑡, iterative and 2𝑛𝑑 order WKB approximations for two interpretations of the FW equation. Section 11 gives the connecting formulae between regions where the WKB approximation is appropriate. Section 12 extends my approach to include periodic structures. Section 13 details how I evaluated the boundary conditions. Section 14 gives numerical results for examples of tunneling through barriers and resonant diodes.

The appendices are organized as follows: Appendix A analytically justifies my commutation of FW wavefunction operators. Appendix B analytically calculates the reflection at a step in both the Dirac spinor and FW representations for comparison. Appendix C examines the tunneling of a massless fermion through a magnetic delta barrier in the FW representation. Appendix D calculates GFs of the FW equation. Appendix E uncovers a relativistic GF paradox.

Dirac equation and its Foldy-Wouthuysen representation

The Dirac equation for a scalar potential 𝑉 is given in Hamiltonian form by

[ 𝑐𝛼 ⋅ p + 𝛽𝑚𝑐 2 ] Ψ = 𝐻Ψ = (𝐸 + 𝑉 ) Ψ , ( 1
)
𝛾 𝑡 = ( +𝐼 0 0 -𝐼 ) , 𝛾 𝑥,𝑦,𝑧 = ( 0 +𝜎 𝑥,𝑦,𝑧 -𝜎 𝑥,𝑦,𝑧 0 ) , (2) 𝐼 = ( +1 0 0 +1 ) , 𝜎 𝑥 = ( 0 +1 +1 0 ) , ( 3
)
𝜎 𝑦 = ( 0 -𝑖 +𝑖 0 ) , 𝜎 𝑧 = ( +1 0 0 -1 ) , (4)
with 𝛼 𝑥,𝑦,𝑧 = 𝛾 𝑡 𝛾 𝑥,𝑦,𝑧 and 𝛽 = 𝛾 𝑡 .

p = -𝑖ℏ∇ = -𝑖ℏ [ 𝜕 𝜕𝑥 , 𝜕 𝜕𝑦 , 𝜕 𝜕𝑧 ] (5)
is the momentum operator for the electron which is of mass 𝑚 and energy 𝐸. The charge and current density are given by the following well known expressions:

𝜌 = Ψ * Ψ , ( 6
)
𝑗 = Ψ * 𝛼Ψ . ( 7
)
There are two linearly independent solutions of the free particle Dirac equation,

Ψ (±) ↑ = ⎛ ⎜ ⎜ ⎜ ⎝ ∓𝑚𝑐 2 -𝐸 -𝑉 0 0 𝑐 ( 𝑝 𝑥 ± 𝑖𝑝 𝑦 ) ⎞ ⎟ ⎟ ⎟ ⎠ exp ( 𝑖𝑝 𝑥 𝑥 + 𝑖𝑝 𝑦 𝑦 ℏ ) , (8)
Ψ (±) ↓ = ⎛ ⎜ ⎜ ⎜ ⎝ 0 ∓𝑚𝑐 2 -𝐸 -𝑉 𝑐 ( 𝑝 𝑥 ∓ 𝑖𝑝 𝑦 ) 0 ⎞ ⎟ ⎟ ⎟ ⎠ exp ( 𝑖𝑝 𝑥 𝑥 + 𝑖𝑝 𝑦 𝑦 ℏ ) , (9)
each can be normalised to one electron per unit volume by Eq. [START_REF] Wan | Flexible n-type thermoelectric materials by organic intercalation of layered transition metal dichalcogenide tis 2[END_REF]. By making use of the FW transformation I will decouple the differential equations of the Dirac equation to give four independent identical equations. I will show that by calculating the propagation for only one of these components I may obtain the full GF and transmission.

Under unitary transformation the Dirac equation becomes

𝑒 +𝑖𝑆 𝐻𝑒 -𝑖𝑆 𝑒 +𝑖𝑆 Ψ = (𝐸 + 𝑉 ) 𝑒 +𝑖𝑆 Ψ . ( 10
)
Foldy and Wouthuysen [START_REF] Foldy | On the dirac theory of spin 1/2 particles and its non-relativistic limit[END_REF] found that their rotation

𝑒 ±𝑖𝑆 = cos 𝜃 ± 𝛽𝛼 ⋅ p | p| sin 𝜃 , ( 11
)
transformed the Dirac Hamiltonian into diagonal and antidiagonal parts

𝛽 ( 𝑚𝑐 2 cos (2𝜃) + 𝑐 | p| sin (2𝜃) ) + 𝑐𝛼 ⋅ p ( cos (2𝜃) - 𝑚𝑐 | p| sin (2𝜃) ) = 𝑒 +𝑖𝑆 𝐻𝑒 -𝑖𝑆 . ( 12
)
Subsequently it was chosen [START_REF] Foldy | On the dirac theory of spin 1/2 particles and its non-relativistic limit[END_REF]

cos (2𝜃) = 𝑚𝑐 2 √ 𝑐 2 p2 + 𝑚 2 𝑐 4 , ( 13
)
sin (2𝜃) = 𝑐 | p| √ 𝑐 2 p2 + 𝑚 2 𝑐 4 , ( 14
)
so that by direct substitution of Eq. ( 13) and Eq. ( 14) into Eq. ( 12)

[ 𝛽 √ p2 𝑐 2 + 𝑚 2 𝑐 4 -1 (𝑉 + 𝐸) ] Ψ = 0 . ( 15
)
Eq. [START_REF] Dosch | Kleins paradox[END_REF] shows there are two possibilities for the FW equation

[ ∓ √ p2 𝑐 2 + 𝑚 2 𝑐 4 -(𝑉 + 𝐸) ] Ψ(±) = 0 , (16)
Ψ(-) corresponds to the electron, Ψ(+) to the positron. The rotated wavefunction Ψ is related to Ψ by

[ cos 𝜃 + 𝛽𝛼 ⋅ p | p| sin 𝜃 ] Ψ = Ψ . ( 17
)
Taylor expanding √ p2 𝑐 2 + 𝑚 2 𝑐 4 in Eq. ( 16) gives

[ ∓ - ( 𝑉 + 𝐸 ± 𝑚𝑐 2 )] Ψ(±) = 0 , ( 18
)
where

 = 𝑚𝑐 2 [ 1 2
[ p 𝑚𝑐 ] 2 - 1 8
[ p 𝑚𝑐 ] 4 + 1 16
[ p 𝑚𝑐 ] 6 -... ] . (19)
Eq. ( 16) and Eq. ( 18) are two interpretations of the FW equation. I demonstrate the unsuitability of my FW approach for calculating magnetic barrier tunneling in Appendix C. I discuss the GFs of the FW Eq. ( 18) in Appendix D. I uncover and discuss a GF paradox of the FW Eq. ( 18) in Appendix E.

Relationship between the transmission of the Dirac equation and its Foldy-Wouthuysen representation

Throughout my numerical and analytic results, Section 14 and Appendix B, I will calculate the reflection coefficient 𝑅 to deduce the transmission coefficient 𝑇 as

𝑇 = 1 -𝑅 . ( 20
)
Calculating 𝑅, rather than 𝑇 directly, does not require the normalisation of the wavefunction in both the incident and transmitted regions.

Let Ψ be given by Eq. ( 8) and Eq. ( 9) as

Ψ = 𝑢 exp ( 𝑖𝑝 𝑥 𝑥 + 𝑖𝑝 𝑦 𝑦 ℏ ) , ( 21
)
taking Ψ of the form

Ψ = ũ exp ( 𝑖𝑝 𝑥 𝑥 + 𝑖𝑝 𝑦 𝑦 ℏ ) , (22)
I find by application of Eq. ( 17) to Eq. ( 21)

ũ1 = 𝑢 1 cos 𝜃 + ( 𝑑 𝑥 -𝑖𝑑 𝑦 ) 𝑢 4 sin 𝜃 , ũ2 = 𝑢 2 cos 𝜃 + ( 𝑑 𝑥 + 𝑖𝑑 𝑦 ) 𝑢 3 sin 𝜃 , ũ3 = 𝑢 3 cos 𝜃 - ( 𝑑 𝑥 -𝑖𝑑 𝑦 ) 𝑢 2 sin 𝜃 , ũ4 = 𝑢 4 cos 𝜃 - ( 𝑑 𝑥 + 𝑖𝑑 𝑦 ) 𝑢 1 sin 𝜃 , (23)
where d = p∕| p|.

Eq. ( 21), Eq. ( 22), and Eq. [START_REF] Masir | Wavevector filtering through single-layer and bilayer graphene with magnetic barrier structures[END_REF] show Ψ in exponential planar form which is useful for calculating the fermion propagation through homogeneous space.

The incident (𝑖) and reflected (𝑟) waves for planar structures in the Dirac equation description are related to the reflection coefficient in the well known way:

𝑅 = (Ψ ⋅ Ψ * ) (𝑟) (Ψ ⋅ Ψ * ) (𝑖) . ( 24
)
For the two linearly independent solutions of the Dirac equation, Eq. ( 8) and Eq. ( 9), Eq. ( 24) gives rise to two degenerate expressions for the reflection coefficient:

𝑅 ↑ = ( | | Ψ 1 | | 2 + | | Ψ 4 | |
Since the components of Ψ are degenerate I am only required to evaluate the propagation of a fermion by the FW Eq. ( 18) once when calculating 𝑅 and 𝑇 .

Boundary Conditions of the Foldy-Wouthuysen equation at a sharp step

Consider a sharp step in potential at 𝑥 = 𝑎. I will integrate the FW Eq. ( 18) through the boundary, normal to the boundary

lim (𝑎 + -𝑎 -)→0 ∫ 𝑎 + 𝑎 - [ ∓ - ( 𝑉 + 𝐸 ± 𝑚𝑐 2 )] Ψ(±) 𝑑𝑥 𝑐ℏ = 0 . ( 31
)
In this section I introduce the operator  which commutes with 𝑐 p𝑥 and is defined by

 Ψ = -𝑐 p𝑥 Ψ = -𝑐 p𝑥  Ψ . ( 32
)
I give a discussion of Eq. (32) in Appendix A. Since

lim (𝑎 + -𝑎 -)→0 ∫ 𝑎 + 𝑎 - ( 𝑉 + 𝐸 ± 𝑚𝑐 2 ) Ψ(±) 𝑑𝑥 𝑐ℏ = 0 , ( 33
)
due to the function integrated being finite but the integration range tending to zero, I have

[ 𝑖 Ψ] 𝑎 + 𝑎 - = lim (𝑎 + -𝑎 -)→0 ∫ 𝑎 + 𝑎 -  Ψ 𝑑𝑥 𝑐ℏ = 0 . ( 34
)
I can understand Eq. (34) when I consider that

 Ψ = 𝑖𝑐ℏ 𝜕 (  Ψ) 𝜕𝑥 . ( 35
)
Eq. (34) shows that Ψ is continuous with respect to the operation of .

In order to evaluate  Ψ for the mathematical description of the boundary conditions note

±𝑐 p𝑥 Ψ(±) = ( 𝑉 + 𝐸 ± 𝑚𝑐 2 ) Ψ(±) = ±𝑐𝑝 𝑥  Ψ(±) , (36) therefore  Ψ(±) = ± 𝑉 + 𝐸 ± 𝑚𝑐 2 𝑐𝑝 𝑥 Ψ(±) = ∓𝛼 (±) Ψ(±) . ( 37
)
For my second boundary condition I note that pΨ always gives a finite observed value, therefore Ψ is continuous everywhere.

My boundary conditions are in stark contrast to the continuity conditions of the Klein-Gordon equation, in that case continuity is with respect to momentum and wavefunction. I verify my boundary conditions numerically and analytically in Section 14 and Appendix B.

Transfer matrices for a square potential barrier

Inside a layer thickness 𝐷 the transfer matrix of Ψ is given by

⎛ ⎜ ⎜ ⎜ ⎝ exp ( - 𝑖𝐷𝑝 𝑥 ℏ ) 0 0 exp ( + 𝑖𝐷𝑝 𝑥 ℏ ) ⎞ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ ã1 b1 ⎞ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎝ ã0 b0 ⎞ ⎟ ⎟ ⎟ ⎠ , ( 38
)
where ã0 and ã1 denote forward traveling FW wave parts and b0 and b1 denote backward travelling FW wave parts, 0 denotes on the left, 1 denotes on the right. The fermion is refracted at the angle 𝜙 to the layer boundary normal, where

𝑝 𝑥 = 𝑝 cos 𝜙 . ( 39
)
For a boundary at a sharp step, applying the wavefunction continuity condition set out in Section 4 gives

ã0 + b0 = ã1 + b1 . ( 40
)
Secondly, applying wavefunction continuity under the operation of , derived in Section 4, to Eq. ( 40) gives

𝛼 0 ã0 -𝛼 0 b0 = 𝛼 1 ã1 -𝛼 1 b1 . ( 41
)
Eq. ( 40) and Eq. ( 41) can be written in matrix form as

1 2 ⎛ ⎜ ⎜ ⎜ ⎝ ( 1 + 𝛼 1 𝛼 0 ) ( 1 - 𝛼 1 𝛼 0 ) ( 1 - 𝛼 1 𝛼 0 ) ( 1 + 𝛼 1 𝛼 0 ) ⎞ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ ã1 b1 ⎞ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎝ ã0 b0 ⎞ ⎟ ⎟ ⎟ ⎠ . ( 42
)
In the event of tunneling, wave modes become an exponentially decaying and growing set. Transfer matrix multiplication combines these large and small values, this process is numerically unstable and inaccurate. Instead I suggest using the method of Refs. [START_REF] Ko | Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals[END_REF][START_REF] Tikhodeev | Quasiguided modes and optical properties of photonic crystal slabs[END_REF] to combine my transfer matrices. The method of Refs. [START_REF] Ko | Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals[END_REF][START_REF] Tikhodeev | Quasiguided modes and optical properties of photonic crystal slabs[END_REF] addresses these instability issues by separating the exponentially growing and decaying terms.

Transfer matrix for a delta potential barrier

Integrating Eq. ( 18), with 𝑉 (𝑥) = 𝑔𝛿(𝑥) and with respect to 𝑑𝑥∕𝑐ℏ, across

[ 0 -, 0 + ] gives [ ∓𝑖 ( ã(±) + b(±) )] 0 + 0 - = 𝑔 𝑐ℏ ( ã(±) + b(±) ) | | | |𝑥=0 . (43)
Continuity of the wavefunction gives

ã0 + b0 = ã1 + b1 . ( 44
)
Eq. ( 43) and Eq. ( 44) can be written in the following matrix form

⎛ ⎜ ⎜ ⎜ ⎝ ( 1 - 𝑖𝑔 2𝑐ℏ𝛼 𝑘 ) ( - 𝑖𝑔 2𝑐ℏ𝛼 𝑘 ) ( + 𝑖𝑔 2𝑐ℏ𝛼 𝑘 ) ( 1 + 𝑖𝑔 2𝑐ℏ𝛼 𝑘 ) ⎞ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎝ ã1 b1 ⎞ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎝ ã0 b0 ⎞ ⎟ ⎟ ⎟ ⎠ , ( 45
)
where I define ã0 , ã1 , b0 , and b1 in the same way as Section 5.

𝟏𝐬𝐭 order relativistic WKB approximation

Consider the following result from Appendix D derived from the correct fermion interpretation of the FW equation in terms of ,

𝑇 (𝑘) = ( 2𝛼 𝑘 𝛼 𝑞 ) 2 ( 𝛼 2 𝑞 -𝛼 2 𝑘 ) 2 sin 2 ( 2𝑚 * 𝑞𝑎 ℏ ) + ( 2𝛼 𝑘 𝛼 𝑞 ) 2 (46)
for a rectangular barrier given by

𝑉 (𝑥) = { 𝑉 for |𝑥| < 𝑎 , 0 for |𝑥| > 𝑎 . ( 47
)
Eq. ( 46) can be approximated as

𝑇 (𝑘) ≈ ( 2𝛼 𝑘 𝛼 𝑞 𝛼 2 𝑘 -𝛼 2 𝑞 ) 2 exp ( - 4𝑖𝑚 * 𝑞𝑎 ℏ ) ( 48
)
by assuming exponential wavefunction decay within the barrier. Taking logarithms of Eq. ( 48), the non-logarithmic term dominates the logarithmic term so that

log 𝑒 𝑇 (𝑘) ≈ - 4𝑖𝑚 * 𝑞𝑎 ℏ . ( 49
)
Assuming independence of transmission events, the total transmission probability for crossing an overall barrier is

𝑇 ≈ ∞ ∏ 𝑖=1 𝑇 𝑖 , ( 50
)
log 𝑒 ∞ ∏ 𝑖=1 𝑇 𝑖 ≈ ∞ ∑ 𝑖=1 - 2𝑖𝑚 * 𝑖 𝑞 𝑖 Δ𝑥 ℏ = - 2𝑖 ℏ ∫ 𝑝 𝑥 𝑑𝑥 , ( 51
)
where the 𝑇 𝑖 are the transmission probabilities of the individual barriers. I see from Eq. ( 51) that inside a smoothly varying layer the FW wavefunction is approximately given by

Ψ ≈ exp ( ± 𝑖 ℏ ∫ 𝑝 𝑥 𝑑𝑥 ) 𝑁 ∏ 𝑖=1 ( 2𝛼 𝑘 𝛼 𝑞 𝛼 2 𝑘 -𝛼 2 𝑞 ) 𝑖 .
(52)

Iterative relativistic WKB approximation

Consider a wavefunction of the form

Ψ = exp ( 𝑖Φ ℏ ) , ( 53
)
then the boson FW interpretation Eq. ( 16) becomes

⎡ ⎢ ⎢ ⎣ √ 𝑐 2 ( 𝜕Φ 𝜕𝑥 ) 2 -𝑐 2 𝑖ℏ 𝜕 2 Φ 𝜕𝑥 2 + 𝑚 2 𝑐 4 -(𝑉 + 𝐸) ⎤ ⎥ ⎥ ⎦ Ψ = 0 . ( 54
)
Eq. ( 54) can be exactly rewritten as

𝑐 2 ( 𝜕Φ 𝜕𝑥 ) 2 -𝑐 2 𝑖ℏ 𝜕 2 Φ 𝜕𝑥 2 + 𝑚 2 𝑐 4 = (𝑉 + 𝐸) 2 . ( 55
)
I now make use of (𝐸 + 𝑉 ) 2 = 𝑚 2 𝑐 4 + 𝑝 2 𝑐 2 in Eq. ( 55) and find that as expected

p2 𝑥 Ψ = 𝑝 2 𝑥 Ψ , 𝜕Φ 𝜕𝑥 = √ 𝑝 2 𝑥 + 𝑖ℏ 𝜕 2 Φ 𝜕𝑥 2 . ( 56
)
I know from Section 7

± 𝜕Φ 𝜕𝑥 ≈ 𝑝 𝑥 , ( 57
)
suggesting a next approximation on from Eq. ( 52) is

Φ ≈ ± ∫ √ 𝑝 2 𝑥 ± 𝑖ℏ 𝜕𝑝 𝑥 𝜕𝑥 𝑑𝑥 . ( 58
)
Eq. ( 58) follows from the integration of Eq. ( 56). This procedure can be iterated to an 𝑛 𝑡ℎ order WKB approximation in the boson FW interpretation.

𝟐𝐧𝐝 order relativistic WKB approximation

Expanding Φ (𝑥) in powers of ℏ

Φ (𝑥) = Φ 0 (𝑥) + ℏΦ 1 (𝑥) + ℏ 2 Φ 2 (𝑥) + ... , (59)
then substituting Eq. (59) into Eq. ( 56) gives

( Φ ′ 0 + ℏΦ ′ 1 + ... ) 2 -𝑖ℏ ( Φ ′′ 0 + ℏΦ ′′ 1 + ... ) = 𝑝 2 𝑥 . (60)
Equating terms in powers of ℏ, in Eq. ( 60), I see

±Φ ′ 0 = 𝑝 𝑥 , 2Φ ′ 0 Φ ′ 1 -𝑖Φ ′′ 0 = 0 , ( 61
)
so that

Φ 0 = ± ∫ 𝑝 𝑥 𝑑𝑥 , Φ 1 = 𝑖 2 log 𝑒 Φ ′ 0 . ( 62
)
Eqs. (62), Eq. ( 59), and Eq. ( 53) give

Ψ ≈ exp ( ± 𝑖 ℏ ∫ 𝑝 𝑥 𝑑𝑥 ) √ 𝑝 𝑥 . ( 63
)
This result is identical to the Klein-Gordon WKB approximation because on the way I approximated FW Eq. ( 18) with Eq. ( 16).

Discussion of fermion and boson WKB approximations

When I made my WKB approximation in Section 7, from what I will show in Section 14 and Appendix B to be the correct form Eq. ( 18) of the FW equation, I saw in the nonexponential factor of Eq. ( 48) appearances of 𝑝 𝑖 changed to appearances of 𝛼 𝑖 going from boson to fermion descriptions. I have noticed, that for tunneling through slowly varying potentials, Eq. (48), Eq. (50) and Eq. ( 63) give the implication:

𝑁 ∏ 𝑖=1 ( 2𝑘𝑞 𝑘 2 -𝑞 2 ) 2 𝑖 ≈ 𝑝 1 𝑝 𝑁 ⟹ 𝑁 ∏ 𝑖=1 ( 2𝛼 𝑘 𝛼 𝑞 𝛼 2 𝑘 -𝛼 2 𝑞 ) 2 𝑖 ≈ 𝛼 (±) 𝑝 1 𝛼 (±) 𝑝 𝑁 . ( 64
)
Eq. (64) suggests that the WKB approximation for fermions, in the correct FW representation, will take the form

Ψ ≈ exp ( ± 𝑖 ℏ ∫ 𝑝 𝑥 𝑑𝑥 ) √ 𝛼 𝑝 𝑥 . ( 65
)
Going from Eq. (63) to Eq. (65) I swapped an appearance of 𝑝 𝑖 to ±𝛼 (±) 𝑖 , from boson to fermion descriptions, as suggested by Eq. (48), Eq. (50) and Eq. (63).

Connection formulae

It is suggested by Section 8 that for the WKB approximation to be appropriate

| | 𝑝 𝑥 | | 2 ≫ | | | | ℏ 𝜕𝑝 𝑥 𝜕𝑥 | | | | . ( 66
)
Eq. (66) suggests that when 𝑝 𝑥 ≈ 0 the WKB method fails. I need a connection formula between the regions where 𝑝 2 𝑥 ≠ 0.

I assume that when Eq. (66) fails

𝑚𝑐 ≫ | | 𝑝 𝑥 | | , ( 67
)
so that I may treat the connecting region with the 𝑐 = ∞ non-relativistic limit of the FW Eq. ( 18). The wavefunction in the connecting region is then given by the Airy function solutions to the Schrödinger equation. This approach leads to the two connected solutions for Ψ between the classical and non-classical regions: Classical region 𝐩 𝟐 > 𝟎 and 𝐱 < 𝐱 𝟏

𝐴 1 exp ( + 𝑖 ℏ ∫ 𝑥 𝑥 1 |𝑝| 𝑑𝑥 ′ ) √ |𝑝| +𝐵 1 exp ( - 𝑖 ℏ ∫ 𝑥 𝑥 1 |𝑝| 𝑑𝑥 ′ ) √ |𝑝| , (68)
Non-classical region 𝐩 𝟐 < 𝟎 and 𝐱 > 𝐱 𝟏

𝐴 2 exp ( - 1 ℏ ∫ 𝑥 𝑥 1 |𝑝| 𝑑𝑥 ′ ) √ |𝑝| +𝐵 2 exp ( + 1 ℏ ∫ 𝑥 𝑥 1 |𝑝| 𝑑𝑥 ′ ) √ |𝑝| , ( 69
)
where

2𝐴 2 = -𝑖 ( 𝐴 1 + 𝑖𝐵 1 ) exp ( + 𝑖𝜋∕4 ) (70)
and

𝐵 2 = +𝑖 ( 𝐴 1 -𝑖𝐵 1 ) exp ( -𝑖𝜋∕4 ) . ( 71
)
In Eq. ( 68) and Eq. (69) I have used the asymptotic forms of the Airy functions and taken 𝑝 = 0 to be at 𝑥 1 . The wavefunctions Eq. ( 68) and Eq. ( 69) are in the WKB form derived for relativistic fermions in Section 7 to Section 10, therefore the solutions in both regions can be extended to relativistic energies.

Extention to periodic structures

Consider the periodic potential

𝑉 (𝑥) = ∑ 𝐺 𝑉 𝐺 exp ( + 𝑖𝐺𝑥 ℏ ) . ( 72
)
Eq. ( 72) is a Fourier expansion of the potential, where 𝐺 = 2𝜋𝑛∕𝑎 and 𝑛 is positive and negative integers. The wavefunction follows from Bloch's theorem for periodic crystals

Ψ(±) = exp ( + 𝑖𝑘𝑥 ℏ ) ∑ 𝑔 𝐴 𝑔 exp ( - 𝑖𝑔𝑥 ℏ ) , ( 73
)
where 𝑔 = 2𝜋𝑚∕𝑎 and 𝑚 is positive and negative integers. Substituting Eq. (72) and Eq. ( 73) into

[ ∓ - ( 𝐸 ± 𝑚𝑐 2 + 𝑉 )] Ψ(±) = 0 (74)
and multiplying through by exp (𝑖(𝐺 ′ -𝑘)𝑥∕ℏ), where 𝐺 ′ = 2𝜋𝑛 ′ ∕𝑎, then integrating 𝑥 over one period 𝑎, I find that the first, second, and third terms are only non-zero for 𝑔 = 𝐺 ′ while the fourth term is only non-zero for 𝑔-𝐺 = 𝐺 ′ . Therefore, with rearrangement

∑ 𝑔 ( 𝛿 𝑔𝐺 ′ 𝑊 𝑔 -𝑉 𝑔-𝐺 ′ ) 𝐴 𝑔 = ( 𝐸 ± 𝑚𝑐 2 ) 𝐴 𝐺 ′ , ( 75
)
where

𝑊 𝑔 = - ( 𝛼 (±) 𝑘 -𝛼 (±) 𝑔 ) (𝑐𝑘 -𝑐𝑔) . ( 76
)
The eigenvalue problem described by Eq. ( 75) can be solved for varying 𝑘 to give the eigenvalues

( 𝐸 ± 𝑚𝑐 2 )
. The relationship between 𝑘 and 𝐸 defines the band structure.

For planar structures of period 𝐷 I have by Bloch's theorem

Ψ (𝑥 + 𝐷) = exp ( 𝑖𝐾𝐷 ℏ ) Ψ (𝑥) , ( 77
)
which in terms of transfer matrices can be written as

𝑀 Ψ = exp ( 𝑖𝐾𝐷 ℏ ) Ψ = 𝜆 Ψ ⟹ |𝑀 -𝜆𝐼| = 0 , ( 78
)
where

𝑀 = ( 𝑀 11 𝑀 12 𝑀 21 𝑀 22 ) . ( 79
)
For fermions to propagate through the periodic structure 𝐾 must be real, therefore by Eqs. ( 78)

-1 ≤ cos ( 𝐾𝐷 ℏ ) = 𝑀 11 + 𝑀 22 2 ≤ +1 . ( 80
)
Band-gaps arise in 1D periodic potentials if the condition in Eq. ( 80) is not met.

Calculation of transfer matrix parameters 13.1. Calculation of the angle of refraction

To match the Dirac spinors of Eq. ( 8) and Eq. ( 9) either side of a potential step at 𝑥 = 0 it must be that

𝑝 (𝑖) 𝑦 | 𝑥=0 = 𝑝 (𝑡) 𝑦 | 𝑥=0 , (81)
therefore, since

p = x𝑝 cos 𝜙 + ŷ𝑝 sin 𝜙 , (82)
I arrive at Snell's law for the relativistic fermion sin 𝜙 (𝑖) sin

𝜙 (𝑡) = 𝑝 (𝑡) 𝑝 (𝑖) . ( 83
)
I do not analytically verify non-normal incident relativistic tunneling and transmission in this article.

Tunneling between layers of varying electron mass

In free space by special relativity

𝐸 2 = 𝑚 2 𝑐 4 ( 1 + ( 𝑘 𝑐 ) 2 ) , ( 84
)
inside a layer by special relativity

(𝐸 + 𝑉 ) 2 = 𝑚 * 2 𝑐 * 4 ( 1 + ( 𝑞 𝑐 * ) 2 ) , ( 85
)
where 𝑚 * is the effective mass of the electron in the layer and 𝑐 * is the corresponding effective speed of light. Comparing Eq. (84) and Eq. ( 85), I find

𝑞 2 𝑘 2 = ( 𝑚𝑐 𝑚 * 𝑐 * ) 2 + 𝑚𝑐 2 𝑚 * 𝑐 * 2 2𝑔(𝑘∕𝑐)𝑉 𝑚 * 𝑘 2 + ( 𝑉 𝑚 * 𝑐 * 𝑘 ) 2 + + ( 𝑚𝑐 2 -𝑚 * 𝑐 * 2 ) ( 𝑚𝑐 2 + 𝑚 * 𝑐 * 2 ) (𝑚 * 𝑐 * ) 2 1 𝑘 2 . ( 86
)
In the last term of Eq. (86) I used the difference of two squares to improve numerical accuracy. When |𝑘| < 𝑐 the function 𝑔(𝑘∕𝑐) is given by the Taylor expansion

𝑔 (𝑘∕𝑐) = √ 1 + ( 𝑘 𝑐 ) 2 = 1 + (𝑘∕𝑐) 2 2 - (𝑘∕𝑐) 4 8 + ... ( 87
)
and when |𝑘| > 𝑐 the function 𝑔(𝑘∕𝑐) is given by the Taylor expansion

𝑔 (𝑘∕𝑐) = ( 𝑘 𝑐 ) ( 1 + (𝑐∕𝑘) 2 2 - (𝑐∕𝑘) 4 8 + ... ) . ( 88
)
Computationally I add smallest terms together first for improved numerical accuracy. In terms of 𝑔(𝑘∕𝑐) and the 𝑥component of electron momentum 𝑚 * 𝑞 𝑥 I calculate the parameter 𝛼 (±) 𝑞 to be

𝛼 (±) 𝑞 = - 𝑚𝑐 2 𝑔 (𝑘∕𝑐) + 𝑉 ± 𝑚 * 𝑐 * 2 𝑐 * 𝑚 * 𝑞 𝑥 . ( 89
)

Tunneling into a layer where electron mass vanishes

In free space the fermion momentum dispersion is given by Eq. (84) whereas inside the layer electron mass has vanished and therefore

𝐸 + 𝑉 = 𝑞𝑐 * , ( 90
)
where 𝑞 is the fermion momentum in the layer, and 𝑐 * is the speed of light in the layer. Comparing Eq. (84) and Eq. ( 90), I find

𝑞𝑐 * = 𝑚𝑐 2 𝑔 (𝑘∕𝑐) + 𝑉 (91)
and

𝛼 𝑞 = - 𝑞 𝑞 𝑥 . ( 92
)

Results

This section is restricted to quantum tunneling. For the structures I investigate in this section I introduce a length scale 𝑎 given in terms of electron mass 𝑚, the speed of light 𝑐, and ℏ In Appendix B I investigate reflection of electrons from a sharp step in potential. In Appendix B I find exact analytic agreement for 𝑅 between the FW and Dirac spinor representations.

𝑎 = 5 ℏ 𝑚𝑐 . ( 93
The striking feature of all my results is that I see exact analytic agreement between 𝑇 and 𝑅 calculated in the FW and Dirac spinor representations for all momentum 𝑝, even when |𝑝| > 𝑚𝑐. This is a significant finding since when |𝑝| > 𝑚𝑐 the Taylor expansion of  is a divergent series. I think that conservation of energy is restricting my evaluation of  on Ψ(±) to the particular values required to meet the physical boundary conditions for the Dirac Hamiltonian.

I model a resonant-tunneling diode to compare my WKB approximations of Section 9 and Section 10 as the final example in this section. For my numerical computations I use Python 3.8.

Tunneling through a barrier

The system under consideration is a potential barrier described by

𝑉 (𝑥) = - { 0 for |𝑥| > 𝑎 , 3𝑚𝑐 2 for |𝑥| < 𝑎 . ( 94
)
Since 𝑉 is negative, the momentum of the electron inside the barrier becomes imaginary when the following condition is met:

-𝑚𝑐 2 < 𝐸 + 𝑉 < +𝑚𝑐 2 , ( 95
)
then the electron is quantum tunneling. The analytic transmission for Eq. ( 94) is derived in Ap-pendix D as:

𝑇 (𝑘) = 4𝜅 2 4𝜅 2 + ( 1 -𝜅 2 ) 2 sin 2 ( 2𝑚𝑞𝑎 ℏ ) , ( 96
)
where I take 𝜅 to be 𝜅 𝑓 in the FW representation, and 𝜅 𝑑 in the Dirac spinor representation. For the derivation of Appendix D in the FW representation

𝜅 (±) 𝑓 = 𝛼 (±) 𝑘 𝛼 (±) 𝑞 = 𝑞 𝑥 𝑘 𝑥 𝐸 ± 𝑚𝑐 2 𝐸 ± 𝑚𝑐 2 + 𝑉 . ( 97
)
It was shown by the authors of Refs. [START_REF] Calogeracos | History and physics of the klein paradox[END_REF][START_REF] Dosch | Kleins paradox[END_REF] by deriving 𝑇 in the Dirac spinors representation of quantum tunneling

𝜅 (±) 𝑑 = 𝑞 𝑥 𝑘 𝑥 𝐸 ± 𝑚𝑐 2 𝐸 ± 𝑚𝑐 2 + 𝑉 , ( 98
)
which is in exact analytic agreement with 𝜅 (±) 𝑓 , as expected. The numerical results demonstrating relativistic tunneling of electrons are shown in Fig. 1. For comparison I also show the transmission of tunneling bosons with identical mass to the electron.

Fig. 1(b) shows the absolute error calculated as the difference between 𝑇 evaluated using the Taylor expansions 𝑔(𝑘∕𝑐) in 𝜅 or the Python 3.8 𝑐𝑚𝑎𝑡ℎ module square root function.

The region of slow convergence around |𝑘∕𝑐| = 1 in Fig. 1(b) is an artifact of the point in 𝑔(𝑘∕𝑐) about which I have taken my Taylor expansions.

Tunneling through a barrier with vanishing electron mass inside the barrier

In this subsection I almost repeat the demonstration of Section 14.1 except that now inside the barrier the electron mass is strictly vanishing.

Since 𝑚 = 0 when |𝑥| < 𝑎 I must re-evaluate 𝜅 𝑑 for Eq. (96):

𝜅 (±) 𝑑 = 𝑞 𝑥 𝑝 𝑥 𝐸 ± 𝑚𝑐 2 𝐸 + 𝑉 , ( 99
)
which is in exact analytic agreement with 𝜅 (±) 𝑓 , as expected. The numerical results demonstrating tunneling of relativistic electrons are shown in Fig. 2. For comparison I also show the transmission of tunneling bosons with identical mass to the electron.

In Fig. 2(b) the absolute error is calculated as the difference between 𝑇 evaluated using the Taylor expansions 𝑔(𝑘∕𝑐) in 𝜅 or the Python 3.8 𝑐𝑚𝑎𝑡ℎ module square root function.

The region of slow convergence around |𝑘∕𝑐| = 1 in Fig. 2 Transmission as a function of bias potential Δ𝑉 for a series of left hand side electron incident energies 𝐸 (𝑖) ∕𝑚𝑐 2 0.400, 0.425, 0.450, 0.475 as labelled, with Γ 𝑖 taken to be ±𝛼 (±) 𝑖 . (c) Difference in transmission, as a function of bias potential Δ𝑉 , between using ±𝛼 (±) 𝑖 or 𝑝 𝑖 for Γ 𝑖 .

Transmission through a resonant-tunneling diode using the WKB approximation

I calculate the transmission through a resonant diode under a range of bias potentials Δ𝑉 . I show a series of potentials describing increasing Δ𝑉 in Fig. 3(a). The transmission as a function of Δ𝑉 is given in Fig. 3(b) for a series of incident energies. I calculate transmission with the two different WKB approximations, derived in Section 9 and Section 10, for comparison in Fig. 3(c).

The resonant-tunneling diode that I consider includes a planar cavity with a smoothly varying potential. For the wavefunction to traverse the cavity the employment of WKB transfer matrices developed in Section 9 and Section 10 are required:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ exp ( - 𝑖 ℏ ∫ 𝑥 1 𝑥 0 𝑝𝑑𝑥 ) √ Γ 0 ∕Γ 1 0 0 exp ( + 𝑖 ℏ ∫ 𝑥 1 𝑥 0 𝑝𝑑𝑥 ) √ Γ 1 ∕Γ 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ã1 b1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ã0 b0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ( 100
)
Γ 𝑖 is 𝑝 𝑖 or ±𝛼 (±)
𝑖 depending on whether I use FW Eq. ( 16) or Eq. ( 18) inside the cavity. The subscript 0 denotes evaluation at 𝑥 0 = -𝑎∕2 the left hand side of the smoothly varying region for which the transfer matrix is applied, 1 is at 𝑥 1 = +𝑎∕2 the right hand side of the smoothly varying region. I define ã0 , ã1 , b0 , and b1 in the same way as Section 5.

Since I have established in Section 14.1 and Section 14.2 that the Taylor series for 𝑔(𝑘∕𝑐) gives convergence to the square root function of the Python 3.8 𝑐𝑚𝑎𝑡ℎ module, I dispense with the Taylor series and use the 𝑐𝑚𝑎𝑡ℎ square root function to calculate the results in this subsection.

I see in Fig. 3(b) and (c) that as the energy of the incident electrons moves to higher values the first resonance of the diode moves towards lower values of Δ𝑉 , as expected. I observe in Fig. 3(b) negative transmission as expected according to the Klein paradox [START_REF] Calogeracos | History and physics of the klein paradox[END_REF][START_REF] Klein | Die reflexion von elektronen an einem potentialsprung nach der relativistischen dynamik von dirac[END_REF].

In Fig. 3(b) and (c) I see no qualitative difference between my two WKB approximations, showing that exponential decay is the dominant factor in the cavity for this example.

Conclusion

In this work I have developed Dirac fermion transfer matrix methods, for all energies, in the FW representations. I have also developed WKB approximations, to all orders, and discussed the validity of the approximations. I have introduced connection formulae between regions of real and imaginary fermion momentum.

I have shown the extension of my method to 2D periodic structures for band engineering [START_REF] Barbier | Extra dirac points in the energy spectrum for superlattices on single-layer graphene[END_REF][START_REF] Barbier | Dirac and kleingordon particles in one-dimensional periodic potentials[END_REF][START_REF] Arovas | Dirac spectrum in piecewise constant one-dimensional (1d) potentials[END_REF].

I have alerted the reader to the limits of my methods by revealing a GF paradox that occurs when transforming between Dirac and FW representations.

I have verified my methods for massive and effectively massless fermions of all momentum by analytic comparison with the Dirac spinor representation of fermion tunneling. I have shown that the FW representation is an exact description of Dirac fermions and is not restricted to semirelativistic energies as first envisioned [START_REF] Foldy | On the dirac theory of spin 1/2 particles and its non-relativistic limit[END_REF].

Acknowledgements

This research was not funded by any research council grants, if you want to help support this research please make a donation: https://gofund.me/34294631. Thank you in advance for whatever you donate.

Future developments will involve the inclusion of RSE perturbation theory [START_REF] Muljarov | Brillouin-wigner perturbation theory in open electromagnetic systems[END_REF][START_REF] Doost | Resonant-state expansion applied to three-dimensional open optical systems[END_REF][START_REF] Armitage | Resonantstate expansion applied to planar waveguides[END_REF][START_REF] Doost | Resonant-state-expansion born approximation with a correct eigen-mode normalisation[END_REF][START_REF] Doost | Resonant-state-expansion born approximation for waveguides with dispersion[END_REF]. I plan to extend the derivation of the normalization which I made in Ref. [START_REF] Doost | Resonant-state expansion applied to three-dimensional open optical systems[END_REF] to Dirac fermions described by the FW Eq. ( 18). I acknowledge that as a sub-sequence to my rigorous normalization derivations in Ref. [START_REF] Doost | Resonant-state expansion applied to three-dimensional open optical systems[END_REF], E. A. Muljarov incorporated his zero frequency mode discovery into the generalized normalization. I acknowledge the first of the RSE waveguide articles, which I reference, was predominantly the work of E. A. Muljarov.

Declaration of competing interests

I declare that I have no competing financial interests or personal relationships that could have influenced the work reported in this article.

A. Discussion of commutation

In this appendix I will provide a discussion of

 = -𝑐 p𝑥 = -𝑐 p𝑥  . ( 101
)
Consider the form of the operator

  = 𝑚𝑐 2 [ 1 2
[ p 𝑚𝑐 ] 2 - 1 8
[ p 𝑚𝑐 ] 4 + 1 16
[ p 𝑚𝑐 ] 6 -... ] , (102)
where

1 2 [ p 𝑚𝑐 ] 2 = - ℏ 2 2𝑚 2 𝑐 2 [ 𝑑 2 𝑑𝑥 2 + 𝑑 2 𝑑𝑦 2 ] , (103) 1 8
[ p 𝑚𝑐 ] 4 = ℏ 4 8𝑚 4 𝑐 4 [ 𝑑 4 𝑑𝑥 4 + 2 𝑑 4 𝑑𝑦 2 𝑑𝑥 2 + 𝑑 4 𝑑𝑦 4 ] , (104)
and the series of polynomial differential operators continues up to 𝑛 𝑡ℎ order. It is clear that I can write

𝑑 𝑛+𝑚 𝑑𝑥 𝑛 𝑑𝑦 𝑚 = 𝑑 𝑑𝑥 𝑑 𝑛+𝑚-1 𝑑𝑥 𝑛-1 𝑑𝑦 𝑚 = 𝑑 𝑛+𝑚-1 𝑑𝑥 𝑛-1 𝑑𝑦 𝑚 𝑑 𝑑𝑥 . ( 105
)
However it is also the case that In the case described by Eq. (109) I cannot extract 𝑑∕𝑑𝑥 from either side of the operator 𝑑 𝑛 ∕𝑑𝑦 𝑛 . Eq. ( 101) is true when I assume that  acts on a function of not just 𝑦 but also 𝑥.

𝑑 𝑛 𝑑𝑦 𝑛 𝑓 (𝑥, 𝑦) = 𝑑 𝑑𝑥 𝑑𝑥 𝑑𝑦 𝑑 𝑛-1 𝑓 𝑑𝑦 𝑛-1 = 𝑑 𝑛-1 𝑑𝑦 𝑛-1

B. Reflection at a step potential in Foldy-Wouthuysen and Dirac spinor representations for comparison

Consider a fermion incident on a step at 𝑥 = 0. The wavefunction for 𝑥 < 0 is

( 𝐸 ± 𝑚𝑐 2 -𝑐𝑝 𝑥 ) exp ( + 𝑖𝑝 𝑥 𝑥 ℏ ) +𝐵 ( 𝐸 ± 𝑚𝑐 2 +𝑐𝑝 𝑥 ) exp ( - 𝑖𝑝 𝑥 𝑥 ℏ ) , (110)
and for 𝑥 > 0

𝐹 ( 𝐸 + 𝑉 ± 𝑚𝑐 * 2 -𝑐 * 𝑞 𝑥 ) exp ( + 𝑖𝑞 𝑥 𝑥 ℏ ) . ( 111
)
By continuity of wavefunction at 𝑥 = 0 I have

( 1 -𝐵 ) 𝑐𝑝 𝑥 = 𝐹 𝑐 * 𝑞 𝑥 , (112)
( 1 + 𝐵 )( 𝐸 ± 𝑚𝑐 2 ) = 𝐹 ( 𝐸 + 𝑉 ± 𝑚𝑐 * 2 ) . ( 113
)
Dividing Eq. (112) by Eq. ( 113) gives

1 -𝐵 1 + 𝐵 = 𝑞 𝑥 𝑐 * 𝑝 𝑥 𝑐 𝐸 ± 𝑚𝑐 2 𝐸 + 𝑉 ± 𝑚𝑐 * 2 = 𝜅 (±) 𝑑 . ( 114
)
Now I turn to the FW representation. The corresponding wavefunction for 𝑥 < 0 is exp

( + 𝑖𝑝 𝑥 𝑥 ℏ ) + 𝐵 exp ( - 𝑖𝑝 𝑥 𝑥 ℏ ) , ( 115
)
and for 𝑥 > 0 𝐹 exp

( + 𝑖𝑞 𝑥 𝑥 ℏ ) . ( 116
)
Applying my boundary condition at 𝑥 = 0, continuity of wavefunction and continuity of wavefunction with respect to , I obtain

( 1 -𝐵 ) 𝛼 (±) 0 = 𝐹 𝛼 (±) 1 , ( 117
)
1 + 𝐵 = 𝐹 . ( 118
)
Dividing Eq. ( 117) by Eq. ( 118) gives

1 -𝐵 1 + 𝐵 = 𝛼 (±) 1 𝛼 (±) 0 = 𝑝 𝑥 𝑐 𝑞 𝑥 𝑐 * 𝐸 + 𝑉 ± 𝑚𝑐 * 2 𝐸 ± 𝑚𝑐 2 = 𝜅 (±) 𝑓 . ( 119
)
In Section 3 I derived 𝑅 = |𝐵| 2 , therefore

𝑅 = | | | | 1 -𝜅 1 + 𝜅 | | | | 2 , ( 120
)
where I use 𝜅 (±) 𝑑 for the Dirac equation and 𝜅 (±) 𝑓 for the FW equation. Substituting either 𝜅 (±) 𝑓 or 𝜅 (±) 𝑑 into Eq. (120) gives analytically identical 𝑇 and 𝑅 between the Dirac and FW representation, as expected. I also calculated 𝜅 (±) 𝑑 in the case of 𝑚 vanishing for 𝑥 > 0 to find exact analytic agreement in 𝑇 and 𝑅 between the Dirac and FW representations, as expected.

𝜅 (±) 𝑑 corresponds to 𝛼 (±) , and is the mathematical consequence of there being two forms for the FW wavefunction Eq. ( 18). The FW wavefunction describes the electron Ψ(-) or positron Ψ(+) [START_REF] Foldy | On the dirac theory of spin 1/2 particles and its non-relativistic limit[END_REF].

I also calculate, for useful comparison, the transmission of bosons with the following Klein-Gordon result

𝜅 𝑘𝑔 = 𝑞 𝑥 𝑐 * 𝑝 𝑥 𝑐 . (121)

C. Examination of a massless fermion tunneling through a magnetic delta barrier in the Foldy-Wouthuysen representation

Consider a massless fermion tunneling through a magnetic delta barrier 𝐵 = 𝐵 0 𝛿 (𝑥) ẑ, where 𝐵 = ∇ × 𝐴. The barrier is described by 𝐴 = 0 except for 𝐴 = 𝐴 𝑦 ŷ ≠ 0 when 𝑥 > 0. The Dirac spinor wavefunction is a solution of

[𝑐𝛼 ⋅ ( p -𝐴)] Ψ = 𝐸Ψ . ( 122
)
Using the approach of Section 2, Eq. ( 122) can be transformed to

[ ∓𝑐 √ ( p -𝐴) 2 ] Ψ(±) = 𝐸 Ψ(±) . ( 123
)
Taylor expanding Eq. ( 123) around p = 0 gives

[ ∓ - ( 𝐸 ± 𝑐𝐴 𝑦 )] Ψ(±) = 0 , ( 124
)
where  is the corresponding linear differential operator. Using the approach of Section 4 I calculate, that at the barrier, the wavefunction is continuous and also continuous with respect to , where I define

 Ψ(±) = ± 𝐸 ± 𝑐𝐴 𝑦 𝑐𝑝 𝑥 Ψ(±) . ( 125
)
I now return to the Dirac spinor representation. In the region 𝐴 = 𝐴 𝑦 ŷ ≠ 0, the general solution for p = ±𝑞 𝑥 x is given by [START_REF] Masir | Directiondependent tunneling through nanostructured magnetic barriers in graphene[END_REF][START_REF] Masir | Wavevector filtering through single-layer and bilayer graphene with magnetic barrier structures[END_REF][START_REF] Masir | Magnetic kronig-penney model for dirac electrons in single-layer graphene[END_REF]

Ψ = ( +𝐸 ±𝑐𝑞 𝑥 -𝑖𝑐𝐴 𝑦 ) exp ( ± 𝑖𝑞 𝑥 𝑥 ℏ ) , ( 126
)
where

𝑐𝑞 𝑥 = √ 𝐸 2 -𝑐 2 𝐴 2 𝑦 .
The reflection is calculated by matching wavefunctions at the step in vector potential, similar to Appendix C for a step in electrostatic potential.

Making use of my boundary conditions I find the reflection coefficient 𝑅 takes the form of Eq. (120), with

𝜅 𝑑 = 𝑐 * 𝑐 𝑞 𝑥 -𝑖𝐴 𝑦 𝑝 𝑥 (127)
for the Dirac spinor representation and

𝜅 𝑓 = 𝑐𝑝 𝑥 𝑐 * 𝑞 𝑥 𝐸 ± 𝑐 * 𝐴 𝑦 𝐸 ( 128
)
for the FW representation. p = 𝑝 𝑥 x = (𝐸∕𝑐) x when 𝐴 = 0.

The conflict between Eq. (127) and Eq. ( 128) shows that 𝑅 calculated in the FW representation does not reproduce 𝑅 calculated in the Dirac spinor representation, FW wavefunctions cannot be used for the calculation of tunneling through magnetic barriers.

D. Analytic Green's functions in 1-dimension

Consider [ ∓ - ( 𝑉 + 𝐸 ± 𝑚𝑐 2 )] 𝐺 (±) ( 𝑥, 𝑥 ′ ) = 𝛿 ( 𝑥 -𝑥 ′ ) . (129)
Let Ψ𝐿 and Ψ𝑅 be the solutions of Eq. ( 18) which separately satisfy the boundary conditions at 𝑥 = -𝑎 and 𝑥 = +𝑎. The GF of Eq. (129) for the case of a 1D barrier can be written as

𝐺 (±) ( 𝑥, 𝑥 ′ ) = Ψ(±) 𝐿 ( 𝑥 < ) Ψ(±) 𝑅 ( 𝑥 > ) 𝑊 (±) , ( 130
)
where 𝑥 < = min (𝑥, 𝑥 ′ ) and 𝑥 > = max (𝑥, 𝑥 ′ ) and

𝑊 (±) = ∓ Ψ(±) 𝐿 𝑖𝑐ℏ Ψ(±) 𝑅 ± Ψ(±) 𝑅 𝑖𝑐ℏ Ψ(±) 𝐿 (131)
is the Wronskian, which does not depend on 𝑥.

To prove Eq. (130) and Eq. ( 131), note that by Eq. ( 129) ) .

∓ lim 𝜖→0 ∫ 𝑥 ′ +𝜖 𝑥 ′ -𝜖 𝐺 (±) ( 𝑥, 𝑥 ′ ) 𝑑𝑥 = 1 , ( 132
(139)

The transmission for a barrier embedded in a vacuum is given as the ratio of 𝐺(+𝑎, -𝑎) and the free space GF 𝐺 𝑓 𝑠 (+𝑎, -𝑎): The free space GF can be written as

[ ∓ - ( 𝐸 ± 𝑚𝑐 2 )] 𝐺 (±) ( 𝑥, 𝑥 ′ ) = 𝛿 ( 𝑥 -𝑥 ′ ) , (141)
therefore by symmetry, when 𝑥 ′ = 0,

𝐺 (±) (𝑥, 0) = 𝐴 (±) ⎧ ⎪ ⎨ ⎪ ⎩ exp ( - 𝑖𝑚𝑘𝑥 ℏ ) for 𝑥 < 0 , exp ( + 𝑖𝑚𝑘𝑥 ℏ ) for 𝑥 > 0 . ( 142
)
Integrating Eq. ( 141) across [ 0 -, 0 + ] gives ∓𝑖𝑐ℏ𝐴 (±)  exp

( + 𝑖𝑚𝑘0 + ℏ ) ±𝑖𝑐ℏ𝐴 (±)  exp ( - 𝑖𝑚𝑘0 - ℏ ) = 1 (143)
and letting 0 + → 0 and 0 -→ 0, I find 152) that in 1D free space (𝑉 = 0):

𝐴 (±) = 1 2𝑖𝑐ℏ𝛼 (±) 𝑘 . ( 144
𝑐 2 ℏ 2 𝐾 ( 𝑥, 𝑥 ′ ) = - ℏ exp ( 𝑖𝑝 | | 𝑥 -𝑥 ′| | ℏ ) 2𝑖𝑝 , ( 153
)
in 2D free space:

𝑐 2 ℏ 2 𝐾 ( 𝜌, 𝜌 ′ ) = 𝑖 4
𝐻 (1) 0

( 𝑝 | | 𝜌 -𝜌 ′| | ℏ ) , ( 154
)
and in 3D free space:

𝑐 2 ℏ 2 𝐾 ( 𝑟, 𝑟 ′ ) = exp ( 𝑖𝑝 | | 𝑟 -𝑟 ′| | ℏ ) 4𝜋 |𝑟 -𝑟 ′ | . ( 155)

)Figure 1 :

 1 Figure 1: (Color online) (a) Transmission through the potential barrier Eq. (94) as a function of incident momentum 𝑝, for fermions (red) and bosons (orange). (b) Convergence of the fermion transmission with increasing number of relativistic correction Taylor expansion terms 0, 5, 10, 20, 40, 80 as labelled.

Figure 2 :

 2 Figure 2: (Color online) Repeat of Fig. 1 except that now electron mass vanishes inside the barrier.

  (b) is an artifact of the point in 𝑔(𝑘∕𝑐) about which I have taken my Taylor expansions.

Figure 3 :

 3 Figure 3: (color online) (a) Potential profile for a representation of a resonant-tunneling diode under a range of bias potentials Δ𝑉 ∕𝑚𝑐 2 0.00, 0.25, 0.50, 0.75, 1.00 as labelled. (b)Transmission as a function of bias potential Δ𝑉 for a series of left hand side electron incident energies 𝐸 (𝑖) ∕𝑚𝑐 2 0.400, 0.425, 0.450, 0.475 as labelled, with Γ 𝑖 taken to be ±𝛼(±) 𝑖 . (c) Difference in transmission, as a function of bias potential Δ𝑉 , between using ±𝛼(±)

  𝑇 (𝑘) = |2𝑐ℏ𝛼 𝑘 𝐺 (+𝑎, -𝑎) | 2 = | | | | 𝐺 (+𝑎, -𝑎) 𝐺 𝑓 𝑠 (+𝑎, -𝑎)

  𝑓 is a function of not just 𝑦 but also 𝑥. I see from Eq. (105) and Eq. (106) that I can extract 𝑑∕𝑑𝑥 from either side of the operator , in these cases.When 𝑓 (𝑥, 𝑦) = 𝑓 (𝑦), then in general

			but
			𝑑 𝑑𝑥	𝑓 (𝑦) = 0 ,			(108)
			therefore in general
			0 ≠	𝑑 𝑛 𝑑𝑦 𝑛 𝑓 (𝑦) ≠	𝑑 𝑛-1 𝑑𝑦 𝑛-1	𝑑𝑥 𝑑𝑦	𝑑 𝑑𝑥	𝑓 (𝑦) = 0 .	(109)
	𝑑𝑥 𝑑𝑦	𝑑𝑓 𝑑𝑥	(106)
	when every term in 𝑑 𝑛 𝑑𝑦 𝑛 𝑓 (𝑦) ≠ 0 ,		(107)

  Ψ2 the left hand solution is given by Ψ2 (+𝑥) = Ψ1 (-𝑥).

	I now calculate the GF for a barrier of width 2𝑎.
	Ψ1 =	⎧ ⎪ ⎪ ⎪ ⎨	exp 𝐶 exp (	+ (	) 𝑖𝑚 * 𝑞𝑥 𝑖𝑚𝑘𝑥 ℏ + ℏ	)	+ 𝐷 exp	(	-	𝑖𝑚 * 𝑞𝑥 ℏ	)	𝑥 > +𝑎 , |𝑥| < 𝑎 ,
		⎪ ⎪ ⎪	𝐴 exp	(	+	𝑖𝑚𝑘𝑥 ℏ	)	+ 𝐵 exp	(	-	𝑖𝑚𝑘𝑥 ℏ	)	𝑥 < -𝑎 .
		⎩
														(135)
	Continuity of Ψ 1 (𝑥) and continuity of Ψ 1 (𝑥) with respect to
	 gives
	𝐶 =	𝛼 𝑞 + 𝛼 𝑘 2𝛼 𝑞	exp	(	+	𝑖𝑚𝑘𝑎 ℏ	)	exp	(	-	𝑖𝑚 * 𝑞𝑎 ℏ	)	, (136)
	𝐷 =	𝛼 𝑞 -𝛼 𝑘 2𝛼 𝑞	exp	(	+	𝑖𝑚𝑘𝑎 ℏ	)	exp	(	+	𝑖𝑚 * 𝑞𝑎 ℏ	)	. (137)
	Therefore the Wronskian is given by
		𝑊 = 2𝑖𝑐ℏ𝛼 𝑞	( 𝐶 2 -𝐷 2 )	.	(138)
														𝑖𝛼 𝑞
	𝑖𝑐ℏ	( 𝛼 2 𝑞 + 𝛼 2 𝑘	)	sin	(	2𝑚 * 𝑞𝑎 ℏ	)	-2𝑐ℏ𝛼 𝑞 𝛼 𝑘 cos	(	2𝑚 * 𝑞𝑎 ℏ
														)
														which implies
														∓𝑖𝑐ℏ𝐺 (±) (	𝑥 + , 𝑥 ′ )	± 𝑖𝑐ℏ𝐺 (±) ( 𝑥 -, 𝑥 ′ )	= 1 , (133)
														then, applying  to Eq. (130) written out explicitly
														𝐺	(	𝑥, 𝑥 ′ )	=	1 𝑊	{	 Ψ𝐿 Ψ𝐿 ( 𝑥 ′ ) ( 𝑥 ) Ψ𝑅  Ψ𝑅 ( ( 𝑥 ′ ) 𝑥 )	for 𝑥 < 𝑥 ′ , for 𝑥 > 𝑥 ′ ,
														(134)
														and substituting into Eq. (133) gives Eq. (131).

I evaluate Eq. (

130

) to give 𝐺(+𝑎, -𝑎):

  reveal a GF paradox occurring when transforming between the FW and Dirac representations.

	Let
	[𝐻 -(𝐸 + 𝑉 )] 	( 𝑟, 𝑟 ′ )	= 𝛿	(	𝑟 -𝑟 ′ ) 1 ,	(145)
	therefore
	𝑒 +𝑖𝑆 [𝐻 -(𝐸 + 𝑉 )] 𝑒 -𝑖𝑆 𝑒 +𝑖𝑆 [ 	(	𝑟, 𝑟 ′ )]	𝑒 -𝑖𝑆
										= 𝑒 +𝑖𝑆 1𝑒 -𝑖𝑆 𝛿	(	𝑟 -𝑟 ′ )	. (146)
	However since
	𝑒 +𝑖𝑆 1𝑒 -𝑖𝑆 𝛿	(	𝑟 -𝑟 ′ )	= 1𝛿	( 𝑟 -𝑟 ′ )	,	(147)
	and since I have already seen in Section 2
	𝑒 +𝑖𝑆 [𝐻 -(𝐸 + 𝑉 )] 𝑒 -𝑖𝑆 =	[	∓ -	(	𝑉 + 𝐸 ± 𝑚𝑐 2 )] 1 ,
											(148)
	I have
	𝑒 +𝑖𝑆 [ 	( 𝑟, 𝑟 ( 𝑟, 𝑟 ′ )	= 1𝛿	( 𝑟 -𝑟 ′ )	,
											(150)
	so that
	[𝐻 + (𝐸 + 𝑉 )] 𝐾	( 𝑟, 𝑟 ′ )	= 	( 𝑟, 𝑟 ′ )	.	(151)
	Multiplying out the brackets in Eq. (150) I obtain
	-𝑐 2 ℏ 2	[ ∇ 2 +	( 𝑝 ℏ	) 2	]	𝐾	( 𝑟, 𝑟 ′ )	= 𝛿	( 𝑟 -𝑟 ′ )	. (152)
	I see from Eq. (

)

E. Green's function paradox

I ′ )] 𝑒 -𝑖𝑆 = 𝐺 (±) ( 𝑟, 𝑟 ′ ) 1 ⟹ 𝑒 -𝑖𝑆 𝐺 (±) ( 𝑟, 𝑟 ′ ) 𝑒 +𝑖𝑆 =  ( 𝑟, 𝑟 ′ ) 1 . (

149

) Eqs. (149) imply that the valid GFs of the Dirac equation are diagonal. Now consider the identity [𝐻 -(𝐸 + 𝑉 )] [𝐻 + (𝐸 + 𝑉 )] 𝐾

Substituting the identity Eq. (151) into Eq. ( 149)

and so

By inspection of Eq. ( 157) and the energy momentum relation

and substituting Eq. (158) into Eq. (149) I also have

Eq. ( 159) is consistent with the relativistic correction to the 1𝑠𝑡 Born approximation in 3D.

Let us test Eq. (158) and Eq. ( 159) by making an analytic comparison between the GF for 1D free space Eq. ( 142) and the GF composed of Eq. ( 159) and Eq. ( 153):

and so I have an apparent paradox.

Let us attempt to resolve the paradox. I have shown in Appendix D that there are two possible forms of the FW GF:

for an electron delta source (-) or a positron source (+). If I add my two expressions 𝐺 (±) (𝑥, 𝑥 ′ ) I find

However if I try to prove that in general

I arrive at a contradiction. The Klein-Gordon GF has the continuity conditions of the Klein-Gordon equation, (𝐸 +𝑉 ) can be a discontinuous function, therefore 2(𝐸 +𝑉 )𝐾 can be discontinuous at boundaries. Therefore Eq. ( 159) implies  may be discontinuous.  has the same boundary conditions as Ψ, Eq. ( 159) may violate the boundary conditions for . Let us now discuss a possible cause of the paradox. To evaluate the FW transformations defined to be

I am required to know the momentum p also at the delta source. To make this calculation of p at the delta source consider Eq. ( 142) from Appendix D,

I see from Eq. ( 165) that

Eq. ( 166) indicates that when 𝑥 = 𝑥 ′ , the momentum p is undefined. Therefore at the delta source the FW transformation is undefined. In 1D it might be impossible to evaluate 𝑒 ±𝑖𝑆 at the delta source.