# A Graph Neural Network with Negative Message Passing for Graph Coloring
††thanks:  This work was supported by an Alexander von Humboldt Professorship
for Artificial Intelligence endowed to Y. Jin by the German Federal Ministry
of Education and Research.

1st Xiangyu Wang  Faculty of Technology
Bielefeld University
33619 Bielefeld, Germany
0000-0002-1832-0905  2nd Xueming Yan  3rd Yaochu Jin ∗ *Corresponding author
School of Information Science and Technology; Faculty of Technology
Guangdong University of Foreign Studies; Bielefeld University
Guangzhou 510000, China; 33619 Bielefeld, Germany
<EMAIL_ADDRESS>Faculty of Technology
Bielefeld University
33619 Bielefeld, Germany
<EMAIL_ADDRESS>
###### Abstract

Graph neural networks have received increased attention over the past years
due to their promising ability to handle graph-structured data, which can be
found in many real-world problems such as recommended systems and drug
synthesis. Most existing research focuses on using graph neural networks to
solve homophilous problems, but little attention has been paid to heterophily-
type problems. In this paper, we propose a graph network model for graph
coloring, which is a class of representative heterophilous problems. Different
from the conventional graph networks, we introduce negative message passing
into the proposed graph neural network for more effective information exchange
in handling graph coloring problems. Moreover, a new loss function taking into
account the self-information of the nodes is suggested to accelerate the
learning process. Experimental studies are carried out to compare the proposed
graph model with five state-of-the-art algorithms on ten publicly available
graph coloring problems and one real-world application. Numerical results
demonstrate the effectiveness of the proposed graph neural network.

###### Index Terms:

graph neural networks, graph coloring, negative message passing, self-
information, aggregator

##  I Introduction

Different types of data, such as traditional unordered data, time series
structured data, and graph-structured data may be encountered in solving real-
world optimization problems. Graph-structured data contains rich relationship
information between the attributes, making it more challenging to effectively
learn the knowledge in the data using traditional machine learning models.
Recently, graph neural networks (GNNs) have become extremely popular in the
machine learning community, thanks to their strong ability to capture
relational information in the data. Many different types of GNNs have been
proposed, and they can be usually categorized according to different
aggregation methods, such as graph convolutional network (GCN) [1], GraphSAGE
[2], graph attention network (GAT) [3], among others. Since GNN enables the
nodes to aggregate their neighbors’ information, it becomes a powerful tool
for solving problems in social networking [4], bioinformatics [5], and
community detection [6], to name a few. Node classification is often
considered as bi- or multi-class classification problems by inputting the
features of nodes and edges and outputting each node’s probability belonging
to different classes. During the optimization process, each node combines
information of its own and its neighbors to generate embedding vectors. By
representing node and edge features in a higher dimensional space, embedding
vectors are used to solve problems by downstream machine learning tasks in a
non-autoregression or autoregression way [7].

However, most of the optimization problems mentioned above are continuous and
homophilous. That is, two nodes between one edge tend to be very similar,
i.e., have similar embedding vectors. These homophilous problems described as
graphs have many real-world applications, and have been studied extensively
[8, 9, 10]. On the contrary, some problems contain heterophily property [11]
where the connected nodes have as different embeddings as possible. For
example, graph coloring problems (GCPs) are a type of classical heterophilous
problems because they are defined to have different colors between the
connected nodes. GCPs have attracted increased research interest since many
real-world applications can be formulated as graph coloring problems, such as
arranging timetables and schedules, managing air traffic flow, and many others
[12]. In fact, GCPs, like most combinatorial optimization problems (COPs), are
NP-hard problems, which require highly intensive computational costs to obtain
the exact optimal solutions. Luckily, some approaches have been proposed to
find approximate optimal solutions of COPs with the help of GNNs within an
acceptable period of computation time [13, 14, 15]. For example, Liu et al.
[16] proposed two supervised residual gated GCNs to directly predict the
entire Pareto set for multi-objective facility location problems.

Figure 1: Example solutions to graph coloring with six nodes and seven edges.
(a) An optimal solution for the given problem, where
$\mathcal{X}(\mathcal{G})=3$. (b) The number of total conflicts is zero, but
the number of used colors can be reduced. (c) A solution using only two
colors, while the $\mathcal{X}(\mathcal{G})$ being $3$, and the number of
total conflicts should be minimized.

GNNs have inherent advantages to represent GCPs and solve such graph-
structured problems, as samples (nodes) can aggregate information from their
neighbors. Recently, some efforts have been made to apply GNNs to solve GCPs,
and different structured graph neural networks have been proposed. GNN-GCP
[17] uses a network containing a GNN to predict the chromatic number of a
given graph and uses the generated embedding vectors for clustering to obtain
a color assignment. Li et al. [18] explored several rules for using a GNN to
solve GCPs, and proposed a graph neural network called GDN. PI-GNN [19] is
inspired by a Potts model, and it is shown to perform well by combining with a
novel loss function.

The above algorithms basically follow the classical network structures, which
are proposed to solve homophilous problems, such as node classification and
link prediction [20]. However, these structures may not necessarily be suited
for solving GCPs, and improvements should be made in the structural design to
better adapt to the heterogeneous characteristics of GCPs. For this reason, a
negative message passing strategy is proposed considering the requirements for
GCPs, i.e., two connected nodes should have as different embedding vectors as
possible. Meanwhile, color assignment with no conflict and stable convergence
during the training process are both desired when solving GCPs. Therefore, a
loss function consisting of a utility objective and a convergence objective is
proposed to guarantee the good performance of the proposed neural network and
a stable training process.

Section II presents the preliminaries of this work, including the definition
of graph coloring problems, the aggregation methods of classical GNNs, and the
related work that use GNNs to solve graph coloring problems. In Section III,
the proposed graph network model with negative message passing and a new loss
function is detailed. Section IV describes the experimental results on the
benchmark graph coloring problems and a real-world problem. Ablation studies
and analysis of the computational complexity are also presented. Finally,
conclusions and future work are given in Section V.

##  II Preliminaries

###  II-A Definition of Graph Coloring Problems

We consider an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where
$\mathcal{V}=\\{1,2,\dots,n\\}$ is the node set, and $\mathcal{E}$ is the edge
set connecting two nodes, represented by $(u,v)\in\mathcal{E}$. The adjacency
matrix $A$ of a graph $\mathcal{G}$ also represents the connection between the
nodes. When nodes $i$ and $j$ are connected, $A(i,j)$ and $A(j,i)$ are equal
to 1; otherwise, elements in these two positions of $A$ are equal to 0.

Graph coloring problems [21, 22, 23] aim to minimize the number of conflicts
while using the minimum number of colors. Mathematically, GCPs can be
formulated in two different forms. In the first formulation, GCPs are defined
as a constrained minimization problem as follows:

$\begin{split}&\min\quad k,\\\
&\mathrm{s.t.}\sum_{(u,v)\in\mathcal{E}}f(u,v)=0,\end{split}$ (1)

where $f(u,v)$ represents the clash between neighboring nodes $u$ and $v$, and
$(u,v)\in\mathcal{E}$. If $u$ and $v$ share the same color, $f(u,v)=1$;
otherwise, $f(u,v)=0$. $k$ is the number of colors used in graphs. If $k$
colors can fill a graph with no clash, this graph is called $k$-colorable. The
chromatic number $\mathcal{X}(\mathcal{G})$ is the minimum of $k$, denoting
the optimum number of colors without resulting in any conflicts in
$\mathcal{G}$.

Alternatively, GCPs can also be expressed as a constraint satisfaction problem
with a given number of colors $k$:

$\begin{split}&\min\sum_{(u,v)\in\mathcal{E}}f(u,v),\\\ &\mathrm{s.t.}\quad
k.\end{split}$ (2)

This formulation aims to minimize the clashes in a graph, given that the
number of colors that can be used is $k$.

The optimization problems defined in Eq. (1) and Eq. (2) are slightly
different, which may be suited for different requirements or priorities in
solving the same problem. Take the assignment of taxis to customer requests as
an example. In Eq. (1), the customer requests are the top priority, and a
minimal number of taxis should be found. In this case, the assumption is that
there is a sufficient number of taxis. However, if the number of taxis is
limited during the peak period, the assignment that can satisfy the majority
of customer requests would be preferred. This work will focus on minimizing
the number of total conflicts under a certain given number of colors, i.e.,
the GC problem will be solved according to the formulation in Eq. (2).

An illustrative example consisting of six nodes and seven edges is given in
Fig. 1. In Fig. 1 (a), three colors are used, and no connected neighboring
nodes have the same color, meaning that this is an optimal solution to the
given example. By contrast, as shown in Fig. 1(b), there is no conflict
between the neighboring nodes, but the number of colors used can still be
further minimized. In Fig. 1 (c), on the other hand, only two colors have been
used, and there is one conflict in color. Overall, Fig. 1 (a) provides an
ideal solution, while solutions in Fig. 1 (b) and (c) may also be applicable
in different scenarios.

###  II-B Classical GNNs

The main difference between graph neural networks and other neural networks is
that the nodes (samples) in GNNs can gather information from their neighbors
before being projected to the next layer, due to the existence of edges [24,
25]. Aggregating embeddings from neighbors and combining them with the node’s
embedding are two critical operators in GNNs called aggregation and
combination. Authors in [18] consider GNNs with aggregation and combination
operators as AC-GNNs. They make GNNs powerful in exploiting and revealing
relationships between nodes.

The input to GNNs is usually the feature vector or randomly-generated vector
of each node. With an adjacency matrix, one node generates a new embedding in
the next hidden layer by combining its own and neighbors’ aggregated
embedding. Therefore, embeddings in the first layer contain first-order
neighborhood information and the $k$-th hidden layer captures the $k$-th
neighborhood information. The aggregation and combination operators may differ
in various GNNs based on different purposes and optimization tasks. In the
following, we briefly review some classical and popular aggregation and
combination methods.

GCN [1] considers the equal importance of a node and its neighbor information,
which integrates the aggregation and combination methods. The embedding of the
$k$-th layer is calculated by $H^{k}=\sigma(\hat{A}H^{k-1}W^{k-1})$, where
$\hat{A}=\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}$ is the
normalized adjacency matrix with self connections, $D$ is the degree matrix,
and $W^{k-1}$ is a trainable weight matrix of the $k-1$-th layers. On the
other hand, some GNNs separate the aggregation and combination methods,
assigning different importance to the node embedding and the aggregated
neighborhood embedding. GraphSAGE [2] is one example of this type; the
embedding of node $v$ in the $k$-th hidden layer is obtained by
$h_{v}^{k}\leftarrow\sigma(W^{k-1}\cdot
CONCAT(h_{v}^{k-1},h_{\mathcal{N}(v)}^{k}))$, where
$h_{\mathcal{N}(v)}^{k}\leftarrow AGGREGATE(h^{k-1}_{u\in\mathcal{N}(v)})$.
There are also many other types of aggregation and combination methods
proposed very recently, such as HetGNN [26], DNA [27], non-local graph neural
networks [28], and SAR [29].

###  II-C Related Work

Only sporadic work on solving graph coloring problems with the help of graph
neural networks have been reported, which are based on supervised or
unsupervised learning.

GNN-GCP [17] generates color embeddings and node embeddings with a process of
learning whether the graph is $k$-colorable or not. At first, $2^{15}$
positive and $2^{15}$ negative GCP instances are generated with ground truth
given by a GCP solver. The proposed neural network learns the difference
between the ground truth and the prediction by using the binary cross entropy
as a loss function. The prediction is gotten through a GNN to aggregate
neighbor information, an RNN to update the embeddings, and an MLP to get the
final logit probability. If the network predicts a graph is k-colorable, the
second stage is applied by clustering vertex embeddings using k-means, and
nodes in the same cluster share the same color.

Different from the above work that relies on supervised learning and a set of
pre-solved GCP instances, unsupervised learning has also been adopted to
tackle GCPs by constructing a loss function without requiring the ground
truth. In general, the output of unsupervised learning for GCPs are
probability vectors of the nodes, indicating which color should be assigned to
each node. GDN [18] uses a margin loss function, which minimizes the distance
between a pre-defined margin and a Euclidean distance between node pairs.
Schuetz et al. [19] take advantage of the close relationship between GCPs and
the Potts model so that the partition function of the Potts model can be
converted to the chromatic function of $\mathcal{G}$ mathematically. Besides,
the Potts model only distinguishes whether neighboring spins are in the same
state or not, which is very similar to the definition of GCPs. Consequently, a
loss function is proposed in PI-GNN that minimizes the inner product of node
pairs. Generally, loss functions in unsupervised learning usually aim to
minimize the similarity between the connected nodes in solving heterophilous
problems such as graph coloring problems.

Besides, solving GCPs with heuristic algorithms has also been studied. Tabucol
[30] is a Tabu-based algorithm that defines Tabu moves and a Tabu list
tailored for GCPs. It changes the color (for example, red) of one randomly
selected node into another color (green) to reduce the number of conflicts.
The color set (green, red) will be put into the Tabu list for certain
iterations to restrict this node from changing to red. An evolutionary
algorithm, HybridEA [31] is proposed to modify traditional ways of generating
offspring and use a Tabu-search method as the mutation operator.

##  III The Proposed Graph Neural Network Model

The main difference between heterophilous and homophilous optimization
problems is that connected nodes in shared edges should express differently
rather than similarly. For example, in node classification tasks, neighboring
nodes should be classified into different classes instead of the same class.
Therefore, the embeddings of connected nodes should be as different as
possible, and the first-order neighbors need to pass negative messages to the
node. On the other hand, the second-order neighborhood may contain information
that positively impacts the node. Therefore, inspired by the heterophilous
property of graph coloring problems, we proposed a new AC-GNN, which is called
GNN-1N by mainly focusing on the first-order (1st) negative message passing
strategy.

To better solve GCPs, the proposed algorithm should take into account two
objectives. One is to find a solution without conflict and the other is to
achieve fast and stable convergence. Based on the above two objectives, a loss
function for solving graph coloring problems in an unsupervised way is
proposed.

###  III-A Forward Propagation

The forward propagation of the proposed framework is described in this
subsection. We apply GraphSAGE [2] as a baseline GNN model, and some
modifications specialized for solving GCPs are made. In the original paper of
GraphSAGE, the authors give three aggregation methods: mean aggregator, LSTM
aggregator, and pool aggregator. The embedding $h_{v}^{k}$ of node $v$ in the
$k$-th layer using the mean aggregator is calculated as follows:

$\begin{split}&h_{\mathcal{N}(v)}^{k}=mean\\{h_{u}^{k-1}\\},\forall
u\in\mathcal{N}(v),\\\ &h_{v}^{k}=\sigma\left(W_{self}^{k}\cdot
h_{v}^{k-1}+W_{neigh}^{k}\cdot h^{k}_{\mathcal{N}(v)}\right),\end{split}$ (3)

where $\mathcal{N}(v)$ is the connected node set of node $v$,
$h_{\mathcal{N}(v)}^{k}$ is the aggregated embedding of neighborhood of $v$,
$\sigma$ is the activation operator, $W_{self}^{k}$ and $W_{neigh}^{k}$ is the
learnable weight matrix for the $k$-th layer. In this work, the mean
aggregator is considered for improvement.

At the beginning of forward propagation, the embedding of each node is usually
the feature vector. However, some graphs in GCPs may not have features, so
embeddings are generated randomly. In the first hidden layer, nodes gather the
first-order neighborhood information, which should make negative
contributions. Thus, the embedding of the first layer is obtained by

$\begin{split}&h_{\mathcal{N}(v)}^{1}=mean\\{h_{u}^{0}\\},\forall
u\in\mathcal{N}(v),\\\ &h_{v}^{1}=\sigma\left(W_{self}^{1}\cdot
h_{v}^{0}-\alpha\cdot W_{neigh}^{1}\cdot
h^{1}_{\mathcal{N}(v)}\right),\end{split}$ (4)

where $h_{v}^{0}$ is the randomly generated feature of the input layer of node
$v$, $\alpha$ is the trainable parameter controlling the negative influence of
the neighborhood. Elements in $W_{self}^{1}$ and $W_{neigh}^{1}$ are all
positive values distributed uniformly from 0 to 1, and $\alpha$ is initialized
to be $0.5$.

This strategy is reasonable and natural. For example, we assume that there is
node $v$ and its two neighbors $u_{1}$ and $u_{2}$, whose embeddings are
$[0.8,0.6,0.1]$, $[0.7,0.1,0.1]$, and $[0.5,0.1,0.7]$, respectively. We let
$W_{self}^{1}$ and $W_{neigh}^{1}$ be an identity matrix, and $\alpha=0.5$,
the embedding of node $v$ is $[0.5,0.55,-0.1]$ calculated by Eq. (4). If
embeddings also represent the probability of the assigned color, $v$ and
$u_{1}$ conflict with each other before applying Eq. (4), as these two nodes
both prefer the first color among the three given colors. The assigned color
of node $v$ changes into the second one after applying Eq. (4). On the other
hand, if Eq. (3) is used as an updating strategy in the first hidden layer,
$v$ and $u_{1}$ will remain to be conflicting with each other.

According to the property of GNNs, the second hidden layer can aggregate the
second-order neighborhood information, which may be able to contribute some
helpful positive influence. Therefore, in the second-order layer, the original
mean aggregator (Eq. (3)) of GraphSAGE is used to generate embeddings
$h_{v}^{2}$.

For graph coloring problems, there are two ways to assign colors to nodes.
Firstly, there are $k$ sets containing nodes, and all nodes in one set have
the same color. The second way uses the $k$-length probability vector of each
node, and the node is assigned to the $i$-th color if the probability in the
$i$-th position of the vector is the largest. This work considers GCPs as an
unsupervised classification task, and the second way is used to assign $k$
colors to different nodes. Therefore, the dimension of $h_{v}^{2}$ equals the
color number $k$. The probability of node $v$ can be obtained as follows:

$p_{v}=softmax\\{h_{v}^{2}\\},$ (5)

where $softmax$ is the softmax operator, i.e.,
$p_{v}(j)=\frac{h_{v}^{2}(j)}{\sum_{i=1}^{k}h_{v}^{2}(i)}$. The final color
assigned to $v$ is the color with the highest probability.

###  III-B Loss Function

A loss function is proposed to achieve the utility-based and the convergence-
based goal in an unsupervised way. The former is to minimize the conflicts
between the connected nodes with a given number of colors, while the latter is
to minimize the uncertainty of nodes, which can stabilize convergence.

As no ground truth is available, the utility-based objective function uses the
probability of nodes to reflect the relationship of connected nodes. The main
idea is to maximize the difference or minimize the similarity between node
pairs. Some loss functions have been proposed based on the above idea. Among
them, the loss function inspired by the Potts model performs very well, which
is therefore adopted as the utility-based objective function in this work:

$f_{utility}=\sum_{(u,v)\in\mathcal{E}}p_{v}^{T}\cdot p_{u}.$ (6)

Eq. (6) only aims to minimize the inner product between two probabilities,
which can be considered as minimizing the similarity between two nodes. This
is reasonable for solving heterophilous problems such as GCPs. Note that no
other a priori knowledge is required, making it suitable for solving other
unsupervised learning problems.

Figure 2: The framework of the proposed method, where the first-order
aggregator applies Eq. (4) and the second-order aggregator uses the original
aggregator Eq. (3). Between the first and the second hidden layers, dropout is
employed to avoid getting stuck in local optimums.

In addition to maximizing the difference between the connected nodes, we also
intend to increase the probability of one node being assigned a certain color
to make the learning process more stable. Therefore, we introduce self-
information to construct $f_{conv}$, which is formulated as follows:

$f_{conv}=\sum_{i=1}^{n}p_{i}^{T}\cdot\log p_{i}.$ (7)

Self-information represents the amount of information in an event. If the
value of self-information is large, it means this event contains more
information, indicating a high uncertainty. Otherwise, the uncertainty of this
event is low. In terms of GCPs, high self-information of one node means the
probabilities of being assigned to different colors are approximately equal.
It results in unstable convergence because the color assignment varies greatly
under small weight changes. Therefore, nodes with small self-information hold
more confidence in the current color assignments, which helps stabilize the
convergence.

TABLE I: Numerical results and data information for the COLOR dataset. Graph Name | # Nodes | # Edges | Color Number | Tabucol [30] | HybridEA [31] | GDN [18] | PI-GCN [19] | PI-SAGE [19] | GNN-1N
---|---|---|---|---|---|---|---|---|---
myciel5 | 47 | 236 | 6 | 0 | 0 | 0 | 0 | 0 | 0
myciel6 | 95 | 755 | 7 | 0 | 0 | 0 | 0 | 0 | 0
queen5-5 | 25 | 160 | 7 | 0 | 0 | 0 | 0 | 0 | 0
queen6-6 | 36 | 290 | 7 | 0 | 0 | 4 | 1 | 0 | 0
queen7-7 | 49 | 476 | 7 | 10 | 9 | 15 | 8 | 0 | 0
queen8-8 | 64 | 728 | 9 | 8 | 5 | 7 | 6 | 1 | 1
queen9-9 | 81 | 1056 | 10 | 5 | 6 | 13 | 13 | 1 | 1
queen8-12 | 96 | 1368 | 12 | 10 | 3 | 7 | 10 | 0 | 0
queen11-11 | 121 | 1980 | 11 | 33 | 22 | 33 | 37 | 17 | 13
queen13-13 | 169 | 3328 | 13 | 42 | 37 | 40 | 61 | 26 | 15

By combining the terms in Eq. (6) and Eq. (7), we get the following loss
function:

$\mathop{\min}F=f_{utility}+\lambda f_{conv}.$ (8)

where $\lambda>0$ is a hyperparameter.

###  III-C The Overall Framework

As shown in Fig. 2, the framework mainly consists of the forward propagation
and the optimization process. In the forward propagation, the randomly
generated embedding $h_{i}^{0}$ is assigned to the $i$-th node. If we take
$n_{1}$ as an example, it aggregates its first-order negative neighborhood
embedding in the first hidden layer, and its second-order neighborhood
embedding in the second layer. Between these two hidden layers, the dropout
[32] is applied to prevent the solver from getting stuck in a local optimum.
After $n_{1}$ aggregates the two-hop neighborhood information, $h_{1}^{2}$ is
obtained, followed by the softmax function to get a probability vector
$p_{1}$. $p_{1}$ contains $k$ elements representing the probabilities of
choosing $k$ colors. After getting the probability vectors of nodes, the loss
function $F$ is calculated with two terms, namely $f_{utility}$ and
$f_{conv}$. The AdamW optimizer is used to compute the gradient and update
weights in the graph neural network.

##  IV Numerical Experiments

In this section, experiments on the COLOR dataset [33] are presented at first,
and then an application to taxi scheduling is applied. An ablation study is
given to demonstrate the stabilization ability of the proposed loss function,
and finally, the computational complexity is analyzed.

###  IV-A Experiments on COLOR Dataset

In this experiment, we use the publicly available COLOR dataset to evaluate
the performance of the proposed algorithm and its peer methods. The COLOR
dataset is a classical, widely used graph dataset in the field of graph
coloring problems, where Myciel graphs are based on the Mycielski
transformation and Queens graphs are constructed on $n$ by $n$ chessboard with
$n^{2}$ nodes. More detailed information on graphs can be found in Table 1,
with the number of nodes and edges in each graph. Eq. (2) is optimized in the
following experiments given the color number $k$, which is shown in Table 1.

Five algorithms are chosen as peer algorithms: Tabucol [30], HybridEA [31],
GDN [18], PI-GCN [19], and PI-SAGE [19]. Tabucol and HybridEA are tabu-based
heuristics algorithms. The rest three algorithms, GDN, PI-GCN, and PI-SAGE,
are GNN-based unsupervised algorithms. The above five methods focus on
minimizing conflicts with given numbers of colors. GNN-GCP mainly predicts the
color number of a given graph. Therefore, it is not included in this
comparison. The results in Table I show the conflicts of each graph ($f(u,v)$
in Eq. (6)) found by the proposed GNN-1N and the algorithms under comparison.

The results of Tabucol and HybridEA are taken from [18], with a maximum run
time, i.e., 24 hours per graph, and the results of GDN, PI-GCN, and PI-SAGE
are taken from [19]. For a fair comparison, we use the same maximum number of
iterations ($10^{5}$) to run our methods (GNN-1N) on GPU. Besides, the early
stopping mechanism is applied within $10^{3}$ iterations if the value of the
loss function changes less than $0.001$. The hyperparameters, including the
dimensions of $h_{i}^{k}$, the learning rate $\eta$ in the AdamW optimizer,
the probability of dropout, and $\lambda$ in the loss function $F$ are
optimized in a similar way to that in [19]. All results listed in Table I are
the best coloring results of all methods, from which we can see that the
proposed method performs the best on all graphs, especially on large and dense
graphs, such as queen11-11 and queen13-13. On the contrary, traditional tabu-
based methods and other machine-learning methods cannot find as few conflicts
as GNN-1N does.

Figure 3: The final color assignment of queen13-13 with 169 nodes given by our
method. Only 15 conflicts highlighted in red lines exist out of 3328 edges (in
grey lines) in this figure.

Figure 4: Taxi scheduling problem. (a) The timetable containing departure time
and arrival time of seven customers. (b) Encoding the timetable into a graph
coloring problem. (c) Optimizing the graph coloring problem with our
unsupervised neural network method and decoding it. Under this scenario,
customs and taxis are nodes and colors in graph coloring problems,
respectively.

To gain more insights into the solutions found by our methods, we plot the
color assignment of queen13-13 in Fig. 3. There are 169 nodes in this graph,
and 13 colors should be used to color nodes. There are 3328 edges plotted with
grey lines and 15 conflicted edges highlighted with red lines. The normalized
error rate is $0.45\%=\frac{15}{3328}\times 100\%$, which is relatively small,
indicating that our method has a good ability to find less-conflict color
assignments.

###  IV-B Application

In this section, we take the taxi scheduling problem as an example to show the
ability of GNN-1N to solve graph-structured problems in real life. We give a
simple scenario for taxis to customer requests. Seven customs call a taxi
company to book taxis one day. They all plan to book a taxi during the evening
rush hour from 17:00 to 18:00 and each confirms a time period. The task is to
satisfy all customers’ requests with an available number of taxis, assuming
that only four taxis are available during this peak hour.

To solve this problem, three steps are taken, which are 1) encoding, 2)
optimization, and 3) decoding. The encoding step transfers the given timetable
into a graph. As shown in Fig. 4 (a), the timetable with departure time and
arrival time of seven customers is shown, and each customer is represented by
a node in the graph. As customers with overlapping schedules cannot use the
same taxi, two nodes should share one edge if two customers have time overlap.
For example, the arrival time of customer $u_{1}$ is earlier than the
departure time of $u_{2}$, so $u_{1}$ and $u_{2}$ are not connected. On the
other hand, the first customer and the third customer plan to use a taxi from
17:13 to 17:15. Therefore, $u_{2}$ and $u_{3}$ are connected, as shown in Fig.
4 (b). After obtaining the graph description of the relationship between
customers’ schedules, the generated graph is optimized by GNN-1N with a
specific color number. The color number is the number of available taxis,
which is four here. Figure 4 (c) shows the final color assignments with four
colors after optimization, and no conflict is found in the solution. According
to the color assignment, seven customers are assigned into four groups, that
is, $\\{u_{1},u_{2},u_{7}\\}$, $\\{u_{3}\\}$, $\\{u_{4},u_{6}\\}$, and
$\\{u_{5}\\}$.

Figure 5: Convergence profiles of conflicts trained by Eq. (6) and Eq. (8) on
queen6-6 and queen8-12. Eq. (6) contains $f_{utility}$ only, and Eq. (8) is
summation of $f_{utility}$ and $\lambda f_{conv}$, where $f_{conv}$ is the
self-information. Figure 6: The runtimes (in seconds) of GNN-1N required on
each graph coloring problem for $10^{5}$ iterations.

###  IV-C Ablation Study and Time Complexity Analysis

An ablation study is conducted to show the effectiveness of self-information
term $f_{conv}$ (Eq. (7)) included in the loss function $F$. Figure 5 shows
the conflicts trained by the loss function with and without $f_{conv}$ on
queen6-6 and queen8-12 over $10^{5}$ iterations. To be specific, Eq. (6) is
the $f_{utility}$ proposed in [19], and Eq. (8) is the loss function proposed
in this paper with adding the self-information term. The hyperparameters are
obtained directly from [19], and the $\lambda$ in Eq. (8) is set to be $0.25$.
As we can see in Fig. 5, the conflicts curve obtained by Eq. (6) is unstable
and fluctuates dramatically sometimes. By contrast, the conflicts curve
trained by Eq. (8) decreases smoothly. The curves in Fig. 5 indicate the
stabilization function of self-information, which can be attributed to the
convergence term in Eq. (8).

The runtime (in seconds) is plotted in Fig. 6, which shows the time required
by GNN-1N to run $10^{5}$ iterations on one graph coloring problem. It is
reasonable that the runtimes increase with the number of nodes and edges
increase, because the computational cost is mainly concentrated in the process
of node aggregation and backpropagation. In general, the time complexity of
GNN-1N is similar to PI-GNN [19], as no additional computation is added to the
proposed algorithm.

##  V Conclusion and Future Work

The graph coloring problem is a classical graph-based problem aiming to find a
color assignment using a given number of colors. As the GC problem is an NP-
hard problem, it is almost impossible to obtain a feasible solution in an
acceptable time. Moreover, due to its graph property, such problems cannot be
easily and effectively solved by conventional neural networks. In this work,
we propose an unsupervised graph neural network (GNN-1N) tailored for solving
GCPs, which combines negative message passing with normal message passing to
handle heterophily. Besides, a loss function with the utility-based objective
and convergence-based objective is proposed for unsupervised learning.
Experimental results on public datasets show that GNN-1N outperforms five
state-of-the-art peer algorithms. In addition, a toy real-world application of
graph coloring problems is also given to demonstrate further the effectiveness
of GNN-1N.

Solving graph-based heterogeneous problems with GNNs is still in its infancy.
The following three improvements could be made on the proposed model. First,
we can consider pre-/post-processing to further decrease the number of
conflicts. Second, the dynamic graph coloring problems are worthy of
investigation, as the conditions may change in the real world. Finally, the
fairness of color assignment is an interesting topic to examine. For example,
each color should be used roughly the same number of times while making sure
that there is no conflict in connecting nodes. For the taxi scheduling
problem, each taxi should have a similar number of customers. Therefore,
fairness coloring is of great practical importance.

## References

  * [1] T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,” _arXiv preprint arXiv:1609.02907_ , 2016.
  * [2] W. Hamilton, Z. Ying, and J. Leskovec, “Inductive representation learning on large graphs,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [3] P. Veličković, G. Cucurull, A. Casanova, A. Romero, P. Lio, and Y. Bengio, “Graph attention networks,” _arXiv preprint arXiv:1710.10903_ , 2017.
  * [4] Y. Zhang, S. Gao, J. Pei, and H. Huang, “Improving social network embedding via new second-order continuous graph neural networks,” in _Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_ , 2022, pp. 2515–2523.
  * [5] H. Xue, V. Rajan, and Y. Lin, “Graph coloring via neural networks for haplotype assembly and viral quasispecies reconstruction,” _arXiv preprint arXiv:2210.12158_ , 2022.
  * [6] J. Bruna and X. Li, “Community detection with graph neural networks,” _stat_ , vol. 1050, p. 27, 2017.
  * [7] Y. Peng, B. Choi, and J. Xu, “Graph learning for combinatorial optimization: a survey of state-of-the-art,” _Data Science and Engineering_ , vol. 6, no. 2, pp. 119–141, 2021.
  * [8] W. Fan, Y. Ma, Q. Li, J. Wang, G. Cai, J. Tang, and D. Yin, “A graph neural network framework for social recommendations,” _IEEE Transactions on Knowledge and Data Engineering_ , 2020.
  * [9] S. Wu, F. Sun, W. Zhang, X. Xie, and B. Cui, “Graph neural networks in recommender systems: a survey,” _ACM Computing Surveys_ , vol. 55, no. 5, pp. 1–37, 2022.
  * [10] A. Strokach, D. Becerra, C. Corbi-Verge, A. Perez-Riba, and P. M. Kim, “Fast and flexible protein design using deep graph neural networks,” _Cell systems_ , vol. 11, no. 4, pp. 402–411, 2020.
  * [11] X. Zheng, Y. Liu, S. Pan, M. Zhang, D. Jin, and P. S. Yu, “Graph neural networks for graphs with heterophily: A survey,” _arXiv preprint arXiv:2202.07082_ , 2022.
  * [12] N. Barnier and P. Brisset, “Graph coloring for air traffic flow management,” _Annals of operations research_ , vol. 130, no. 1, pp. 163–178, 2004.
  * [13] C. K. Joshi, T. Laurent, and X. Bresson, “An efficient graph convolutional network technique for the travelling salesman problem,” _arXiv preprint arXiv:1906.01227_ , 2019.
  * [14] M. J. Schuetz, J. K. Brubaker, and H. G. Katzgraber, “Combinatorial optimization with physics-inspired graph neural networks,” _Nature Machine Intelligence_ , vol. 4, no. 4, pp. 367–377, 2022.
  * [15] J. Toenshoff, M. Ritzert, H. Wolf, and M. Grohe, “Graph neural networks for maximum constraint satisfaction,” _Frontiers in artificial intelligence_ , vol. 3, p. 580607, 2021.
  * [16] S. Liu, X. Yan, and Y. Jin, “End-to-end pareto set prediction with graph neural networks for multi-objective facility location,” _arXiv preprint arXiv:2210.15220_ , 2022.
  * [17] H. Lemos, M. Prates, P. Avelar, and L. Lamb, “Graph colouring meets deep learning: Effective graph neural network models for combinatorial problems,” in _2019 IEEE 31st International Conference on Tools with Artificial Intelligence (ICTAI)_. IEEE, 2019, pp. 879–885.
  * [18] W. Li, R. Li, Y. Ma, S. O. Chan, D. Pan, and B. Yu, “Rethinking graph neural networks for the graph coloring problem,” _arXiv preprint arXiv:2208.06975_ , 2022.
  * [19] M. J. Schuetz, J. K. Brubaker, Z. Zhu, and H. G. Katzgraber, “Graph coloring with physics-inspired graph neural networks,” _arXiv preprint arXiv:2202.01606_ , 2022.
  * [20] M. Zhang and Y. Chen, “Link prediction based on graph neural networks,” _Advances in neural information processing systems_ , vol. 31, 2018.
  * [21] T. R. Jensen and B. Toft, _Graph coloring problems_. John Wiley & Sons, 2011.
  * [22] P. M. Pardalos, T. Mavridou, and J. Xue, “The graph coloring problem: A bibliographic survey,” in _Handbook of combinatorial optimization_. Springer, 1998, pp. 1077–1141.
  * [23] M. Demange, T. Ekim, B. Ries, and C. Tanasescu, “On some applications of the selective graph coloring problem,” _European Journal of Operational Research_ , vol. 240, no. 2, pp. 307–314, 2015.
  * [24] J. Zhou, G. Cui, S. Hu, Z. Zhang, C. Yang, Z. Liu, L. Wang, C. Li, and M. Sun, “Graph neural networks: A review of methods and applications,” _AI Open_ , vol. 1, pp. 57–81, 2020.
  * [25] Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and S. Y. Philip, “A comprehensive survey on graph neural networks,” _IEEE transactions on neural networks and learning systems_ , vol. 32, no. 1, pp. 4–24, 2020.
  * [26] C. Zhang, D. Song, C. Huang, A. Swami, and N. V. Chawla, “Heterogeneous graph neural network,” in _Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining_, 2019, pp. 793–803.
  * [27] M. Fey, “Just jump: Dynamic neighborhood aggregation in graph neural networks,” _arXiv preprint arXiv:1904.04849_ , 2019.
  * [28] M. Liu, Z. Wang, and S. Ji, “Non-local graph neural networks,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2021.
  * [29] H. Mostafa, “Sequential aggregation and rematerialization: Distributed full-batch training of graph neural networks on large graphs,” _Proceedings of Machine Learning and Systems_ , vol. 4, pp. 265–275, 2022\.
  * [30] A. Hertz and D. de Werra, “Using tabu search techniques for graph coloring,” _Computing_ , vol. 39, no. 4, pp. 345–351, 1987.
  * [31] P. Galinier and J.-K. Hao, “Hybrid evolutionary algorithms for graph coloring,” _Journal of combinatorial optimization_ , vol. 3, no. 4, pp. 379–397, 1999.
  * [32] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” _The journal of machine learning research_ , vol. 15, no. 1, pp. 1929–1958, 2014.
  * [33] M. Trick, “Color dataset,” _URL_ , 2002.


#  Scalar sector of the Myers and Pospelov model: thermal and size effects

L. H. A. R. Ferreira<EMAIL_ADDRESS>Instituto de Física,
Universidade Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  A. F. Santos
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal de
Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  Carlos M. Reyes<EMAIL_ADDRESS>Centro de Ciencias Exactas, Universidad del Bío-Bío,
Casilla 447, Chillán, Chile

###### Abstract

The scalar sector of the Myers and Pospelov model is considered. This theory
introduces a dimension 5 operator with a preferred four-vector which breaks
Lorentz symmetry. We investigate various applications using the TFD formalism,
a topological field theory that allows the study of thermal and size effects
on an equal footing. In this context, Lorentz-violating corrections to the
Casimir effect and Stefan-Boltzmann law have been calculated.

##  I Introduction

Symmetry is a fundamental element in all systems of nature, and it emerges in
different forms within a system. All physical theories are constructed under a
set of symmetries. What does it mean, in physics, for a system to have some
symmetry? It means it is invariant under a certain type of transformation. As
an example of a physical system constructed under a specific set of
symmetries, let us consider the Standard Model (SM), a well-established theory
describing all elementary particles and their interactions, except for the
gravitational interaction, which is described by General Relativity (GR). The
foundations of the SM are the Lorentz and CPT symmetries. Lorentz symmetry is
a fundamental principle stating that experimental results are independent of
the orientation or boost velocity of the laboratory through space. CPT
symmetry predicts the equality of certain quantities, such as lifetime, mass,
gyromagnetic ratio, and charge-to-mass ratio, for particles and antiparticles.
Although the SM is a successful theory, it is not considered a fundamental
theory because it excludes gravitational interaction and does not provide
consistent answers for certain problems, such as dark matter, matter-
antimatter asymmetry, the hierarchy problem, among others. In the last few
decades, new and extended theories have been constructed to solve these
problems. An important attempt is the development of models that search for a
fundamental theory unifying the SM and GR. In these theories beyond the SM,
tiny violations of Lorentz and CPT symmetries could emerge [1, 2].

To study the violation of Lorentz and CPT symmetries in quantum field theories
and gravity, a comprehensive framework has been constructed. This framework is
called Standard Model Extension (SME) [3, 4]. This framework contains the
known physics of the SM, GR, and all possible operators that break Lorentz
symmetry. It is important to emphasize that Lorentz symmetry is composed of
two parts: observer Lorentz transformations and particle Lorentz
transformations. A change of the frame is characterized by an observer Lorentz
transformation, while a rotation or boost performed on an individual particle
field or laboratory setting with a fixed coordinate frame consists of a
particle Lorentz transformation. The SME is invariant under observer
transformations. Furthermore, the SME is divided into two sectors: (i) the
minimal sector of the Standard Model Extension (mSME), which contains
conventional quantization, hermiticity, gauge invariance, power counting
renormalizability, and positivity of the energy, and (ii) the nonminimal
sector of the SME (nmSME), which is associated with operators of higher
dimensions. In this work, we study the scalar sector of the Myers and Pospelov
model [5]. This model introduces a preferred four-vector $\displaystyle n$
that breaks Lorentz symmetry and couples to a five-dimensional operator. In
the case the preferred vector has time components the theory develops higher-
order time derivatives leading to the increase of degrees of freedom
associated to ghost states.

Several studies have been developed from this model. For example, the
conservation of unitarity at one-loop order using the optical theorem and
focusing on a quartic interaction term has been proved [6]. Additionally, a
perturbative expansion of the scalar sector in the Myers-Pospelov model, up to
second order in the Lorentz-violating parameter and taking into account its
higher-order time derivative character, has been constructed [7], among
others. Here, the topological structure of the Thermo Field Dynamics (TFD)
formalism is used to calculate corrections due to the Myers and Pospelov term
for the Casimir effect and the Stefan-Boltzmann law associated with the
massive scalar field.

The TFD formalism is a real-time thermal quantum field theory [8, 9, 10, 11,
12, 13] that was constructed by considering the statistical average of an
arbitrary operator as the expectation value in a thermal vacuum. In order to
construct the thermal vacuum, the Hilbert space is duplicated and the
Bogoliubov transformation is introduced. TFD is a topological field theory
that allows the investigation of different phenomena on an equal footing. Its
topological structure is given as
$\displaystyle\Gamma_{D}^{d}=(\mathbb{S}^{1})^{d}\times\mathbb{R}^{D-d}$ with
$\displaystyle 1\leq d\leq D$. Here, $\displaystyle D$ represents the space-
time dimensions and $\displaystyle d$ is the number of compactified
dimensions. In this formalism, any set of dimensions of the manifold
$\displaystyle\mathbb{R}^{D}$ can be compactified, where the circumference of
the $\displaystyle nth$ $\displaystyle\mathbb{S}^{1}$ is specified by
$\displaystyle\alpha_{n}$, which is the compactification parameter.
Considering the scalar sector in the Myers-Pospelov model, three different
topologies are investigated in this work. These applications involve
introducing thermal and size effects into the theory on an equal footing. Two
different phenomena, such as the Stefan-Boltzmann law and the Casimir effect,
associated with the Myers-Pospelov model, are then studied. Consequently, the
Lorentz-violating corrections to these effects are calculated and analyzed.

This paper is organized as follows. In Section II, the TFD formalism is
briefly presented. In Section III, the scalar sector of the Myers-Pospelov
model is introduced, and the energy-momentum tensor for this theory is
calculated. To obtain a physical and finite quantity, some renormalization
procedure is carried out. In Section IV, different topologies of the TFD
structure are considered. In Subsections A and B, thermal and size effects are
investigated separately. In Subsection C, both effects are analyzed, leading
to the Casimir effect at finite temperature. In all applications, different
directions of the four-vector $\displaystyle n^{\mu}$ are discussed. In
Section V, some concluding remarks are provided.

##  II Thermo Field Dynamics Formalism

In this section, a brief introduction to TFD formalism is presented. This
formalism is constructed under two main ingredients. The first ingredient
implies that the Hilbert space is doubled to create a thermal Hilbert space,
defined as $\displaystyle{\cal S}_{T}={\cal S}\otimes\tilde{\cal S}$, where
$\displaystyle{\cal S}$ represents the standard Hilbert space and
$\displaystyle\tilde{\cal S}$ is the dual or tilde space. There is a map
between the tilde $\displaystyle\tilde{A_{i}}$ and non-tilde $\displaystyle
A_{i}$ operators that is defined by the tilde (or dual) conjugation rules:

$\displaystyle\displaystyle(A_{i}A_{j})^{\thicksim}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\tilde{A_{i}}\tilde{A_{j}},\quad(\tilde{A_{i}})^{\thicksim}=-\xi
A_{i},$ (1) $\displaystyle\displaystyle(A_{i}^{\dagger})^{\thicksim}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\tilde{A_{i}}^{\dagger},\quad(cA_{i}+A_{j})^{\thicksim}=c^{*}\tilde{A_{i}}+\tilde{A_{j}},$

where $\displaystyle\xi=-1\,(+1)$ for bosons (fermions). The second ingredient
consists of the Bogoliubov transformation, which incorporates size and thermal
effects through a rotation between tilde ($\displaystyle\tilde{\cal S}$) and
non-tilde ($\displaystyle{\cal S}$) spaces. As an example of the use of the
Bogoliubov transformation, let’s consider an arbitrary operator
$\displaystyle{\cal O}$ in the Hilbert space $\displaystyle{\cal S}$ and its
corresponding operator $\displaystyle\tilde{\cal O}$ in the tilde space
$\displaystyle\tilde{\cal S}$. By applying this transformation, we obtain

$\displaystyle\displaystyle\left(\begin{array}[]{cc}{\cal O}(k,\alpha)\\\
\xi\tilde{\cal O}^{\dagger}(k,\alpha)\end{array}\right)={\cal
U}(\alpha)\left(\begin{array}[]{cc}{\cal O}(k)\\\ \xi\tilde{\cal
O}^{\dagger}(k)\end{array}\right),$ (6)

where $\displaystyle{\cal U}(\alpha)$ is the Bogoliubov transformation defined
as

$\displaystyle\displaystyle{\cal
U}(\alpha)=\left(\begin{array}[]{cc}u(\alpha)&-w(\alpha)\\\ \xi
w(\alpha)&u(\alpha)\end{array}\right),$ (9)

with $\displaystyle u^{2}(\alpha)+\xi w^{2}(\alpha)=1$. Here,
$\displaystyle\alpha$ is the compactification parameter defined by
$\displaystyle\alpha=(\alpha_{0},\alpha_{1},\cdots\alpha_{D-1})$, which is
associated with the topological structure of the TFD formalism. Using the
topological structure of this theory, any set of dimensions of the manifold
can be compactified. For example, it is possible to choose a topology where
$\displaystyle\alpha_{0}\equiv\beta$ and
$\displaystyle\alpha_{1},\cdots\alpha_{D-1}=0$, with
$\displaystyle\beta=1/k_{B}T$ and $\displaystyle k_{B}$ being the Boltzmann
constant. In this case, thermal effects are introduced into the theory.
Alternatively, we can take $\displaystyle\alpha_{3}\neq 0$ and other
components equal to zero. This introduces size effects. The main advantage is
the fact that two different effects, thermal and size effects, can be
implemented in the same way.

Since our objective is to calculate some applications using the topological
structure of TFD formalism for the scalar field with Lorentz violation, let’s
investigate how the Green function of the scalar field changes due to the
compactification parameter $\displaystyle\alpha$. The propagator of the scalar
field in the doubled notation of TFD formalism, with corrections due to the
$\displaystyle\alpha$ parameter [9, 11], is given as

$\displaystyle\displaystyle G_{0}^{(AB)}(x-x^{\prime};\alpha)=i\langle
0,\tilde{0}|\tau[\phi^{A}(x;\alpha)\phi^{B}(x^{\prime};\alpha)]|0,\tilde{0}\rangle,$
(10)

where $\displaystyle A$ and $\displaystyle B$ $\displaystyle=1,2$ and
$\displaystyle\tau$ is the time ordering operator. To introduce the
compactification parameter in the field, the Bogoliubov transformation is
used, i.e.,

$\displaystyle\displaystyle\phi(x;\alpha)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle{\cal U}(\alpha)\phi(x){\cal U}^{-1}(\alpha).$ (11)

In a similar way, the thermal vacuum is constructed as
$\displaystyle|0(\alpha)\rangle={\cal U}(\alpha)|0,\tilde{0}\rangle$, with
$\displaystyle|0,\tilde{0}\rangle=|0\rangle\otimes|\tilde{0}\rangle$. Note
that $\displaystyle|0\rangle$ and $\displaystyle|\tilde{0}\rangle$ are the
vacuum states in the Hilbert and dual spaces, respectively. Then the
propagator becomes

$\displaystyle\displaystyle G_{0}^{(AB)}(x-x^{\prime};\alpha)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle i\langle
0(\alpha)|\tau[\phi^{A}(x)\phi^{B}(x^{\prime})]|0(\alpha)\rangle,$ (12)
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
i\int\frac{d^{4}k}{(2\pi)^{4}}e^{-ik(x-x^{\prime})}G_{0}^{(AB)}(k;\alpha),$

where

$\displaystyle\displaystyle G_{0}^{(AB)}(k;\alpha)={\cal
U}^{-1}(\alpha)G_{0}^{(AB)}(k){\cal U}(\alpha),$ (13)

with

$\displaystyle\displaystyle
G_{0}^{(AB)}(k)=\left(\begin{array}[]{cc}G_{0}(k)&0\\\ 0&\xi
G^{*}_{0}(k)\end{array}\right),$ (16)

and $\displaystyle G_{0}(k)=(k^{2}-m^{2}+i\epsilon)^{-1}$, where
$\displaystyle m$ is the scalar field mass. It is important to note that, the
relevant quantities are associated with the non-tilde variables. Then the
physical Green function is given by the component $\displaystyle A=B=1$, i.e.,

$\displaystyle\displaystyle G_{0}^{(11)}(k;\alpha)=G_{0}(k)+\xi
w^{2}(k;\alpha)[G^{*}_{0}(k)-G_{0}(k)],$ (17)

where $\displaystyle w^{2}(k;\alpha)$ is given as

$\displaystyle\displaystyle
w^{2}(k;\alpha)=\sum_{s=1}^{d}\sum_{\\{\sigma_{s}\\}}2^{s-1}\sum_{l_{\sigma_{1}},...,l_{\sigma_{s}}=1}^{\infty}(-\xi)^{s+\sum_{r=1}^{s}l_{\sigma_{r}}}\,\exp\left[{-\sum_{j=1}^{s}\alpha_{\sigma_{j}}l_{\sigma_{j}}k^{\sigma_{j}}}\right].$
(18)

This is the generalized Bogoliubov transformation [14]. Here $\displaystyle d$
is the number of compactified dimensions, $\displaystyle\\{\sigma_{s}\\}$
denotes the set of all combinations with $\displaystyle s$ elements and
$\displaystyle k$ is the 4-momentum.

In the next section, the Lorentz-violating extension of the Myers-Pospelov
model is described. The energy-momentum tensor for this theory, considering
the TFD formalism, is calculated.

##  III The model and the energy-momentum tensor

The Lagrangian describing the Lorentz violating extension Myers-Pospelov in
the scalar sector is given as

$\displaystyle\displaystyle{\cal L}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-m^{2}\phi^{\dagger}\phi+ig\phi^{\dagger}(n\cdot\partial)^{3}\phi,$
(19)

where the third term encodes the breakdown of Lorentz symmetry, the constant
vector $\displaystyle n^{\mu}$ acts as a background field and $\displaystyle
g$ is the coupling constant. It is convenient to use the symmetrized
Lagrangian

$\displaystyle\displaystyle{\cal L}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-m^{2}\phi^{\dagger}\phi+\frac{ig}{2}\left(\phi^{\dagger}(n\cdot\partial)^{3}\phi-((n\cdot\partial)^{3}\phi^{\dagger})\phi\right).$
(20)

To find the equation of motion we use the extended Euler-Lagrange equation,
i.e.,

$\displaystyle\displaystyle 0=\frac{\partial{\cal
L}}{\partial\phi}-\partial_{\mu}\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi)}-\partial_{\nu 1}\partial_{\nu
2}\partial_{\mu}\left(\frac{\partial{\cal L}}{\partial({\partial_{\nu
1}\partial_{\nu 2}\partial_{\mu}\phi})}\right).$ (21)

We have

$\displaystyle\displaystyle\frac{\partial{\cal L}}{\partial\phi}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-m^{2}\phi^{\dagger}-\frac{ig}{2}(\partial\cdot
n)^{3}\phi^{\dagger},$ (22) $\displaystyle\displaystyle\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi)}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\partial^{\mu}\phi^{\dagger},$ (23)
$\displaystyle\displaystyle\frac{\partial{\cal L}}{\partial({\partial_{\nu
1}\partial_{\nu 2}\partial_{\mu}\phi})}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{ig}{2}\phi^{\dagger}n^{\nu 1}n^{\nu
2}n^{\mu}.$ (24)

Then the equation of motion becomes

$\displaystyle\displaystyle-m^{2}\phi^{\dagger}-\Box\phi^{\dagger}-ig(\partial\cdot
n)^{3}\phi^{\dagger}=0.$ (25)

Analogously, by taking the dagger operation we obtain the equation of motion
for the field $\displaystyle\phi$

$\displaystyle\displaystyle-m^{2}\phi-\Box\phi+ig(\partial\cdot n)^{3}\phi=0.$
(26)

In order to investigate some applications using TFD formalism the main
quantity that must be calculated is the energy-momentum tensor associated with
the theory. In the ausence of second-order derivatives, we use the expression
for the energy-momentum tensor [15]

$\displaystyle\displaystyle T^{\mu}_{\alpha}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\delta^{\mu}_{\alpha}{\cal L}+\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi)}\partial_{\alpha}\phi+\partial_{\alpha}\phi^{\dagger}\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi^{\dagger})}+\Biggl{\\{}\partial_{\nu
1}\partial_{\nu 2}\left(\frac{\partial{\cal L}}{\partial(\partial_{\nu
1}\partial_{\nu 2}\partial_{\mu}\phi)}\right)\partial_{\alpha}\phi$ (27)
$\displaystyle\displaystyle-$ $\displaystyle\displaystyle\partial_{\nu
1}\left(\frac{\partial{\cal L}}{\partial(\partial_{\nu 1}\partial_{\nu
2}\partial_{\mu}\phi)}\right)\partial_{\nu
2}\partial_{\alpha}\phi+\left(\frac{\partial{\cal L}}{\partial(\partial_{\nu
1}\partial_{\nu 2}\partial_{\mu}\phi)}\right)\partial_{\nu 1}\partial_{\nu
2}\partial_{\alpha}\phi\Biggl{\\}}$ $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\Biggl{\\{}(\partial_{\alpha}\phi^{\dagger})\partial_{\nu
1}\partial_{\nu 2}\left(\frac{\partial{\cal L}}{\partial(\partial_{\nu
1}\partial_{\nu 2}\partial_{\mu}\phi)}\right)-(\partial_{\nu
2}\partial_{\alpha}\phi^{\dagger})\partial_{\nu 1}\left(\frac{\partial{\cal
L}}{\partial(\partial_{\nu 1}\partial_{\nu 2}\partial_{\mu}\phi)}\right)$
$\displaystyle\displaystyle+$ $\displaystyle\displaystyle\partial_{\nu
1}\partial_{\nu 2}\partial_{\alpha}\phi^{\dagger}\left(\frac{\partial{\cal
L}}{\partial(\partial_{\nu 1}\partial_{\nu
2}\partial_{\mu}\phi)}\right)\Biggl{\\}}.$

Let us write each piece

$\displaystyle\displaystyle\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi)}=\partial^{\mu}\phi^{\dagger},\quad\quad\quad\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi^{\dagger})}=\partial^{\mu}\phi,$ (28)
$\displaystyle\displaystyle\frac{\partial{\cal L}}{\partial(\partial_{\nu
1}\partial_{\nu 2}\partial_{\mu}\phi)}=\frac{ig}{2}\phi^{\dagger}n^{\nu
1}n^{\nu 2}n^{\mu},\quad\quad\quad\frac{\partial{\cal
L}}{\partial(\partial_{\nu 1}\partial_{\nu
2}\partial_{\mu}\phi^{\dagger})}=-\frac{ig}{2}\phi n^{\nu 1}n^{\nu 2}n^{\mu},$
(29) $\displaystyle\displaystyle\partial_{\nu 1}\partial_{\nu
2}\frac{\partial{\cal L}}{\partial(\partial_{\nu 1}\partial_{\nu
2}\partial_{\mu}\phi)}=\frac{ig}{2}n^{\mu}(n\cdot\partial)^{2}\phi^{\dagger},\quad\quad\quad\partial_{\nu
1}\partial_{\nu 2}\frac{\partial{\cal L}}{\partial(\partial_{\nu
1}\partial_{\nu
2}\partial_{\mu}\phi^{\dagger})}=-\frac{ig}{2}n^{\mu}(n\cdot\partial)^{2}\phi,$
(30) $\displaystyle\displaystyle\partial_{\nu 1}\frac{\partial{\cal
L}}{\partial(\partial_{\nu 1}\partial_{\nu
2}\partial_{\mu}\phi)}=\frac{ig}{2}n^{\mu}n^{\nu
2}(n\cdot\partial)\phi^{\dagger},\quad\quad\quad\partial_{\nu
1}\frac{\partial{\cal L}}{\partial(\partial_{\nu 1}\partial_{\nu
2}\partial_{\mu}\phi^{\dagger})}=-\frac{ig}{2}n^{\mu}n^{\nu
2}(n\cdot\partial)\phi.$ (31)

Using these pieces, Eqs. (28), (29), (30) and (31), and the Lagrangian (20),
the energy-momentum tensor is given as

$\displaystyle\displaystyle T^{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\eta^{\mu\nu}\left(\partial^{\lambda}\phi^{\dagger}\partial_{\lambda}\phi-m^{2}\phi^{\dagger}\phi\right)+\partial^{\mu}\phi^{\dagger}\partial^{\nu}\phi+\partial^{\nu}\phi^{\dagger}\partial^{\mu}\phi$
(32) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{ig}{2}\Bigl{[}-\eta^{\mu\nu}\left(\phi^{\dagger}(n\cdot\partial)^{3}\phi-((n\cdot\partial)^{3}\phi^{\dagger})\phi\right)$
$\displaystyle\displaystyle+$ $\displaystyle\displaystyle
n^{\mu}\Bigl{(}(n\cdot\partial)^{2}\phi^{\dagger}\partial^{\nu}\phi-(n\cdot\partial)\phi^{\dagger}(n\cdot\partial)\partial^{\nu}\phi+\phi^{\dagger}(n\cdot\partial)^{2}\partial^{\nu}\phi$
$\displaystyle\displaystyle-$
$\displaystyle\displaystyle\partial^{\nu}\phi^{\dagger}(n\cdot\partial)^{2}\phi+(n\cdot\partial)\partial^{\nu}\phi^{\dagger}(n\cdot\partial)\phi-(n\cdot\partial)^{2}(\partial^{\nu}\phi^{\dagger})\phi\Bigl{)}\Bigl{]}.$

To simplify, let us rewrite the energy-momentum tensor as

$\displaystyle\displaystyle T^{\mu\nu}=T^{\mu\nu}_{LI}+T^{\mu\nu}_{LV},$ (33)

where the Lorentz invariant part is given as

$\displaystyle\displaystyle
T^{\mu\nu}_{LI}=-\eta^{\mu\nu}\left(\partial^{\lambda}\phi^{\dagger}\partial_{\lambda}\phi-m^{2}\phi^{\dagger}\phi\right)+2\partial^{\mu}\phi^{\dagger}\partial^{\nu}\phi$
(34)

and the Lorentz violating part is

$\displaystyle\displaystyle T^{\mu\nu}_{LV}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{ig}{2}\Bigl{[}-\eta^{\mu\nu}\left(\phi^{\dagger}(n\cdot\partial)^{3}\phi-((n\cdot\partial)^{3}\phi^{\dagger})\phi\right)$
(35) $\displaystyle\displaystyle+$ $\displaystyle\displaystyle
n^{\mu}\Bigl{(}(n\cdot\partial)^{2}\phi^{\dagger}\partial^{\nu}\phi-(n\cdot\partial)\phi^{\dagger}(n\cdot\partial)\partial^{\nu}\phi+\phi^{\dagger}(n\cdot\partial)^{2}\partial^{\nu}\phi$
$\displaystyle\displaystyle-$
$\displaystyle\displaystyle\partial^{\nu}\phi^{\dagger}(n\cdot\partial)^{2}\phi+(n\cdot\partial)\partial^{\nu}\phi^{\dagger}(n\cdot\partial)\phi-(n\cdot\partial)^{2}(\partial^{\nu}\phi^{\dagger})\phi\Bigl{)}\Bigl{]}.$

In order to avoid divergences, the energy-momentum tensor is written at
different points in space-time, i.e.,

$\displaystyle\displaystyle T^{\mu\nu}_{LI}(x)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lim_{x^{\prime}\to
x}\tau\Bigl{\\{}-\eta^{\mu\nu}\left(\partial^{\lambda}\phi(x)^{\dagger}\partial^{\prime}_{\lambda}\phi(x^{\prime})-m^{2}\phi(x)^{\dagger}\phi(x^{\prime})\right)+2\partial^{\mu}\phi(x)^{\dagger}\partial^{\prime\nu}\phi(x^{\prime})\Bigl{\\}},$
(36)

where $\displaystyle\tau$ is the ordering operator. In the same form the
Lorentz violating part is written as

$\displaystyle\displaystyle T^{\mu\nu}_{LV}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lim_{x^{\prime}\to
x}\tau\Biggl{\\{}\frac{ig}{2}\Bigl{[}-\eta^{\mu\nu}\left(\phi(x)^{\dagger}(n\cdot\partial^{\prime})^{3}\phi(x^{\prime})-((n\cdot\partial)^{3}\phi(x)^{\dagger})\phi(x^{\prime})\right)$
(37) $\displaystyle\displaystyle+$ $\displaystyle\displaystyle
n^{\mu}\Bigl{(}(n\cdot\partial)^{2}\phi(x)^{\dagger}\partial^{\prime\nu}\phi(x^{\prime})-(n\cdot\partial)\phi(x)^{\dagger}(n\cdot\partial^{\prime})\partial^{\prime\nu}\phi(x^{\prime})+\phi(x)^{\dagger}(n\cdot\partial^{\prime})^{2}\partial^{\prime\nu}\phi(x^{\prime})$
$\displaystyle\displaystyle-$
$\displaystyle\displaystyle\partial^{\nu}\phi(x)^{\dagger}(n\cdot\partial^{\prime})^{2}\phi(x^{\prime})+(n\cdot\partial)\partial^{\nu}\phi(x)^{\dagger}(n\cdot\partial^{\prime})\phi(x^{\prime})-(n\cdot\partial)^{2}(\partial^{\nu}\phi(x)^{\dagger})\phi(x^{\prime})\Bigl{)}\Bigl{]}\Biggl{\\}}.$

To proceed with some applications, let’s use the time ordering operator
$\displaystyle\tau$ and consider the commutation relations in higher order
time derivatives theories.

In principle we have a higher-order theory and for some choices of the
preferred four-vector $\displaystyle n^{\mu}$ we have an increase in the
degrees of freedom. This means that phase space is increased since we also
have a momentum variable associated to velocities or time derivative of the
fields $\displaystyle\dot{\phi}$. To deal with this issue and control the
increase of degrees of freedom, here we apply the perturbative method
developed in [16], which by working at the level of the symplectic structure
of the theory, allows to preserve perturbative degrees of freedom, suppress
the ghost-states modes and produce a positive-definite Hamiltonian. The
perturbative reduction was implemented for the scalar Myers-Pospelov model
through a second order expansion in $\displaystyle g$ of the symplectic two
form $\displaystyle\Omega$ given in [7]. In this context, the quantization of
the Myers-Pospelov (MP) model provides perturbative corrections to the
standard commutation relations at equal time [7], i. e.,

$\displaystyle\displaystyle[\phi(x),\phi^{\dagger}(x^{\prime})]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle-
ig\delta^{3}(\vec{x}-\vec{x^{\prime}}),$ (38)
$\displaystyle\displaystyle[\phi(x),\partial^{\prime\mu}\phi^{\dagger}(x^{\prime})]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
v_{0}^{\mu}(1+3g^{2}E^{2})\delta^{3}({\vec{x}-\vec{x^{\prime}}}),$ (39)
$\displaystyle\displaystyle[\phi^{\dagger}(x),\partial^{\prime\mu}\phi(x^{\prime})]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
v_{0}^{\mu}(1+3g^{2}E^{2})\delta^{3}({\vec{x}-\vec{x^{\prime}}}),$ (40)
$\displaystyle\displaystyle[\partial^{\mu}\phi(x),\partial^{\prime\mu}\phi^{\dagger}(x^{\prime})]$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-2iv_{0}^{\mu}gE^{2}\delta^{3}({\vec{x}-\vec{x^{\prime}}}).$
(41)

where $\displaystyle v_{0}^{\mu}=(1,0,0,0)$ is a time-like vector,
$\displaystyle\partial^{\mu}\theta(x_{0}-x_{0}^{\prime})=v_{0}^{\mu}\delta(x_{0}-x_{0}^{\prime})$,
$\displaystyle g$ is the coupling constant and $\displaystyle
E^{2}=m^{2}-\nabla_{x}^{2}$. Using these ingredients, Eq. (36) can written as

$\displaystyle\displaystyle T^{\mu\nu}_{LI}(x)=\lim_{x^{\prime}\to
x}\Bigl{\\{}\Pi^{\mu\nu}\tau[\phi^{\dagger}(x)\phi(x^{\prime})]+\Theta^{\mu\nu}\delta(x-x^{\prime})\Bigg{\\}}$
(42)

with

$\displaystyle\displaystyle\Pi^{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\eta^{\mu\nu}\left(\partial^{\lambda}\partial^{\prime}_{\lambda}-m^{2}\right)+2\partial^{\mu}\partial^{\prime\nu},$
(43) $\displaystyle\displaystyle\Theta^{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
2(1+3g^{2}E^{2})(\eta^{\mu\nu}+v_{0}^{\mu}v_{0}^{\nu}).$ (44)

In a similar form, after some calculations, Eq. (37) becomes

$\displaystyle\displaystyle T^{\mu\nu}_{LV}(x)=\frac{ig}{2}\lim_{x^{\prime}\to
x}\Bigl{\\{}\Gamma^{\mu\nu}\tau[\phi^{\dagger}(x)\phi(x^{\prime})]+I^{\mu\nu}(x-x^{\prime})\Bigg{\\}}$
(45)

where

$\displaystyle\displaystyle\Gamma^{\mu\nu}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\eta^{\mu\nu}\left((n\cdot\partial^{\prime})^{3}-((n\cdot\partial)^{3}\right)$
(46) $\displaystyle\displaystyle+$ $\displaystyle\displaystyle
n^{\mu}\Bigl{(}(n\cdot\partial)^{2}\partial^{\prime\nu}-(n\cdot\partial)(n\cdot\partial^{\prime})\partial^{\prime\nu}+(n\cdot\partial^{\prime})^{2}\partial^{\prime\nu}$
$\displaystyle\displaystyle-$
$\displaystyle\displaystyle\partial^{\nu}(n\cdot\partial^{\prime})^{2}+(n\cdot\partial)\partial^{\nu}(n\cdot\partial^{\prime})-(n\cdot\partial)^{2}\partial^{\nu}\Bigl{)},$

and

$\displaystyle\displaystyle I^{\mu\nu}(x-x^{\prime})$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle-
ig\eta^{\mu\nu}\delta^{3}(\vec{x}-\vec{x^{\prime}})\left[(n\cdot\partial^{\prime})^{3}-(n\cdot\partial)^{3}\right]\theta(x_{0}-x_{0}^{\prime})$
(47) $\displaystyle\displaystyle+$ $\displaystyle\displaystyle
n^{\mu}(1+3g^{2}E^{2})\Bigl{\\{}\delta^{3}(\vec{x}-\vec{x^{\prime}})\Bigl{[}v_{0}^{\mu}(n\cdot\partial)^{2}+n_{\rho}v_{0}^{\rho}(n\cdot\partial^{\prime})\partial^{\prime\nu}\Bigl{]}$
$\displaystyle\displaystyle-$
$\displaystyle\displaystyle(n\cdot\partial^{\prime})v_{0}^{\nu}\delta^{3}(\vec{x}-\vec{x^{\prime}})(n\cdot\partial)-(n\cdot\partial^{\prime})^{2}v_{0}^{\nu}\delta^{3}(\vec{x}-\vec{x^{\prime}})$
$\displaystyle\displaystyle-$
$\displaystyle\displaystyle(n\cdot\partial)v_{0}^{\nu}\delta^{3}(\vec{x}-\vec{x^{\prime}})(n\cdot\partial^{\prime})+n_{\rho}v_{0}^{\rho}\delta^{3}(\vec{x}-\vec{x^{\prime}})(n\cdot\partial)\partial^{\nu}\Bigl{\\}}\theta(x_{0}-x_{0}^{\prime})$
$\displaystyle\displaystyle+$ $\displaystyle\displaystyle
ign^{\mu}\Bigl{[}(n\cdot\partial)^{2}\delta^{3}(\vec{x}-\vec{x^{\prime}})v_{0}^{\nu}\delta(x_{0}-x_{0}^{\prime})+\delta^{3}(\vec{x}-\vec{x^{\prime}})(n\cdot\partial^{\prime})^{2}\partial^{\prime\nu}\theta(x_{0}-x_{0}^{\prime})$
$\displaystyle\displaystyle-$
$\displaystyle\displaystyle\partial^{\nu}\delta^{3}(\vec{x}-\vec{x^{\prime}})(n\cdot\partial^{\prime})^{2}\theta(x_{0}-x_{0}^{\prime})-\delta^{3}(\vec{x}-\vec{x^{\prime}})(n\cdot\partial)^{2}\partial^{\nu}\theta(x_{0}-x_{0}^{\prime})\Bigl{]}.$

Taking the vacuum expectation value of Eq. (33), we get

$\displaystyle\displaystyle\braket{T^{\mu\nu}(x)}$
$\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\braket{0}{T^{\mu\nu}(x)}{0}$ (48)
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\lim_{x^{\prime}\to
x}{\bigg{\\{}}\Pi^{\mu\nu}\braket{0}{\tau[\phi^{\dagger}(x)\phi(x^{\prime})]}{0}+\Theta^{\mu\nu}\delta(x-x^{\prime})\braket{0}{0}$
$\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{ig}{2}\left(\Gamma^{\mu\nu}\braket{0}{\tau[\phi^{\dagger}(x)\phi(x^{\prime})]}{0}+I^{\mu\nu}(x,x^{\prime})\braket{0}{0}\right){\bigg{\\}}}.$

Using the definition of the propagator for a massive scalar field, i.e.,

$\displaystyle\displaystyle\braket{0}{\tau[\phi^{\dagger}(x)\phi(x^{\prime})]}{0}$
$\displaystyle\displaystyle=iG_{0}(x-x^{\prime})$ (49)

where

$\displaystyle\displaystyle
G_{0}(x-x^{\prime})=-\frac{im}{4\pi^{2}}\frac{K_{1}(m\sqrt{-(x-x^{\prime})^{2}})}{\sqrt{-(x-x^{\prime})^{2}}}$
(50)

with $\displaystyle K_{\nu}(z)$ being the Bessel function, Eq. (48) becomes

$\displaystyle\displaystyle\braket{T^{\mu\nu}(x)}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\lim_{x^{\prime}\to
x}{\bigg{\\{}}i\Pi^{\mu\nu}G_{0}(x-x^{\prime})+\Theta^{\mu\nu}\delta(x-x^{\prime})$
(51) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{ig}{2}\left(i\Gamma^{\mu\nu}G_{0}(x-x^{\prime})+I^{\mu\nu}(x,x^{\prime})\right){\bigg{\\}}}.$

Considering the topological structure of the TFD approach, this equation is
written

$\displaystyle\displaystyle\braket{T^{\mu\nu}(x;\alpha)}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\lim_{x^{\prime}\to
x}{\bigg{\\{}}i\Pi^{\mu\nu}G_{0}^{(ab)}(x-x^{\prime};\alpha)+\Theta^{\mu\nu}\delta(x-x^{\prime})$
(52) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{ig}{2}\left(i\Gamma^{\mu\nu}G_{0}^{(ab)}(x-x^{\prime};\alpha)+I^{\mu\nu}(x,x^{\prime})\right){\bigg{\\}}}.$

where $\displaystyle a,b=1,2$ denoting the doubled notation.

To obtain a physical or finite energy-momentum tensor, the Casimir
prescription is used. Then

$\displaystyle\displaystyle{\cal T}^{\mu\nu(ab)}(x;\alpha)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\braket{T^{\mu\nu(ab)}(x;\alpha)}-\braket{T^{\mu\nu(ab)}(x)}$
(53) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\lim_{x^{\prime}\to
x}{\bigg{\\{}}\left[i\Pi^{\mu\nu}-\frac{g}{2}\Gamma^{\mu\nu}\right]\overline{G}^{(ab)}_{0}(x-x^{\prime};\alpha){\bigg{\\}}},$

with

$\displaystyle\displaystyle\overline{G}^{(ab)}_{0}(x-x^{\prime};\alpha)=G_{0}^{(ab)}(x-x^{\prime};\alpha)-G^{(ab)}_{0}(x-x^{\prime}).$
(54)

In the next section, Eq. (53) is used to investigate thermal and size effects
for different choices of the constant vector $\displaystyle n^{\mu}$, which
lead to the breaking of Lorentz symmetries.

##  IV Thermal and size Applications

To investigate the thermal and size effects associated with a massive scalar
field with higher-order Lorentz symmetry breaking, let us consider the
topological structure of the TFD formalism. Using this approach it is possible
to analyze different phenomena on an equal footing. Here three different
topologies are considered: 1 - The topology
$\displaystyle\Gamma_{4}^{1}=\mathbb{S}^{1}\times\mathbb{R}^{3}$, where
$\displaystyle\alpha=(\beta,0,0,0)$. Here, the time-axis is compactified in
$\displaystyle\mathbb{S}^{1}$, with circumference $\displaystyle\beta$. 2 -
The topology $\displaystyle\Gamma_{4}^{1}$ with
$\displaystyle\alpha=(0,0,0,i2d)$, where the compactification along the
coordinate $\displaystyle z$ is considered. 3 - The topology
$\displaystyle\Gamma_{4}^{2}=\mathbb{S}^{1}\times\mathbb{S}^{1}\times\mathbb{R}^{2}$
with $\displaystyle\alpha=(\beta,0,0,i2d)$ is used. In this case, two axes are
compactified, time and coordinate $\displaystyle z$.

###  IV.1 Thermal effects

To analyze thermal effects the compactification parameter is taken as
$\displaystyle\alpha=(\beta,0,0,0)$. This compactification allows us to
calculate the Stefan-Boltzmann law associated with the Lorentz violation
theory. For such a development, the Bogoliubov transformation is given as

$\displaystyle\displaystyle\omega^{2}(\beta)=\sum^{\infty}_{l_{0}=1}e^{-\beta
k^{0}l_{0}},$ (55)

and the Green function is

$\displaystyle\displaystyle\overline{G}_{0}(x-x^{\prime};\beta)=2\sum^{\infty}_{l_{0}=1}G_{0}(x-x^{\prime}-i\beta
l_{0}v_{0}),$ (56)

where $\displaystyle v^{\mu}_{0}=(1,0,0,0)$.

Now let us analyze the energy-momentum tensor given in Eq. (53) for different
choices of the constant vector $\displaystyle n^{\mu}$ that acts as a
background field.

####  IV.1.1 Time-like constant vector

In this case, the constant vector is chosen as $\displaystyle
n^{\mu}=(1,0,0,0)$. Then, the energy-momentum tensor for
$\displaystyle\mu=\nu=0$ becomes

$\displaystyle\displaystyle{\cal T}^{00(11)}(x;\alpha)=2\lim_{x^{\prime}\to
x}\sum_{l_{0}=1}^{\infty}{\bigg{\\{}}\left[i\Pi^{00}-\frac{g}{2}\Gamma^{00}\right]G_{0}(x-x^{\prime}-i\beta
l_{0}v_{0}){\bigg{\\}}},$ (57)

where

$\displaystyle\displaystyle\Pi^{00}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\partial^{0}\partial^{\prime
0}+\partial^{1}\partial^{\prime 1}+\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime 3}+m^{2},$ (58)
$\displaystyle\displaystyle\Gamma^{00}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 2\partial^{0}\partial^{0}\partial^{\prime
0}-2\partial^{0}\partial^{\prime 0}\partial^{\prime 0}.$ (59)

This expression leads to the Stefan-Boltzmann law associated with the massive
scalar field, i.e.,

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{m^{2}}{\pi^{2}\beta^{2}}\sum^{\infty}_{l_{0}=1}\frac{1}{l_{0}^{2}}\Biggl{\\{}(m\beta
l_{0})K_{1}(m\beta l_{0})+3K_{2}(m\beta l_{0})$ (60)
$\displaystyle\displaystyle+$ $\displaystyle\displaystyle\frac{g}{\beta
l_{0}}\left[(m\beta l_{0})^{2}K_{0}(m\beta l_{0})+5(m\beta l_{0})K_{1}(m\beta
l_{0})+12K_{2}(m\beta l_{0})\right]\Biggl{\\}}.$

It is important to note that for this choice of vector $\displaystyle n^{\mu}$
the Lorentz violating term contributes to the energy associated with the
scalar field.

To be clearer about the correction due to the higher-order Lorentz symmetry
breaking, let us consider the limit of small mass bound, i.e., $\displaystyle
m\to 0$. Then Eq. (60) becomes

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta)$
$\displaystyle\displaystyle=\frac{\pi^{2}}{15\beta^{4}}+\frac{24g\zeta(5)}{\pi^{2}\beta^{5}},$
(61)

where $\displaystyle\zeta(x)$ is the Riemann zeta function. Here has been used
that in such a limit the Bessel function is given as

$\displaystyle\displaystyle K_{\nu}(z)\approx\Gamma(\nu)2^{\nu-1}z^{-\nu}.$
(62)

It is important to note in Eq. (61) that the correction for Lorentz violation
contributes to an increase in energy. Furthermore, the temperature dependence
of the Lorentz-violating component does not follow $\displaystyle T^{4}$, but
rather $\displaystyle T^{5}$, indicating that at high temperatures, this term
can become dominant.

####  IV.1.2 Space-like constant vector

Here, the constant vector $\displaystyle n^{\mu}$ is chosen as $\displaystyle
n^{\mu}=(0,1,0,0)$, $\displaystyle n^{\mu}=(0,0,1,0)$ and $\displaystyle
n^{\mu}=(0,0,0,1)$. Following steps similar to the previous subsection, it is
noted that the higher-order Lorentz violation term does not contribute to
these choices of $\displaystyle n^{\mu}$. Then the Stefan-Boltzmann law is the
standard result associated to the massive scalar field, i.e.,

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{m^{2}}{\pi^{2}\beta^{2}}\sum^{\infty}_{l_{0}=1}\frac{1}{l_{0}^{2}}\Biggl{\\{}(m\beta
l_{0})K_{1}(m\beta l_{0})+3K_{2}(m\beta l_{0})]\Biggl{\\}}.$ (63)

Therefore, the correction due to Lorentz violation depends on the direction of
the constant vector that acts as a background field.

###  IV.2 Size effects

In this section, to investigate the size effects, the compactification
parameter is taken as $\displaystyle\alpha=(0,0,0,i2d)$. This means that the
compactification along the coordinate $\displaystyle z$ is considered, which
allows calculating the Casimir effect at zero temperature associated with the
massive scalar field with higher-order Lorentz symmetry breaking.

For this topology, the Bogoliubov transformation is given as

$\displaystyle\displaystyle\omega^{2}(d)=\sum^{\infty}_{l_{3}=1}e^{-i2dk^{3}l_{3}},$
(64)

and the Green functions is written as

$\displaystyle\displaystyle\overline{G}_{0}(x-x^{\prime};d)=2\sum^{\infty}_{l_{3}=1}G_{0}(x-x^{\prime}-2dl_{3}v_{3}),$
(65)

with $\displaystyle v^{\mu}_{3}=(0,0,0,1)$. In this context, the Casimir
effect at zero temperature will be calculated for different direction of the
constant vector $\displaystyle n^{\mu}$.

####  IV.2.1 Time-like constant vector

Assuming $\displaystyle n^{\mu}=(1,0,0,0)$ the component
$\displaystyle\mu=\nu=0$ of the energy-momentum tensor becomes

$\displaystyle\displaystyle{\cal T}^{00(11)}(x;d)=2\lim_{x^{\prime}\to
x}\sum_{l_{3}=1}^{\infty}{\bigg{\\{}}\left[i\Pi^{00}-\frac{g}{2}\Gamma^{00}\right]G_{0}(x-x^{\prime}-2dl_{3}v_{3}){\bigg{\\}}},$
(66)

using Eqs. (58) and (59), it is obtained that

$\displaystyle\displaystyle{\cal
T}^{00(11)}(d)=-\frac{m^{2}}{4\pi^{2}d^{2}}\sum_{l_{3}=1}^{\infty}\frac{K_{2}(2mdl_{3})}{l_{3}^{2}}.$
(67)

This is the Casimir energy for the massive scalar field, where we can see that
the violation of Lorentz symmetries has no influence on its value. In the
following let us calculate the Casimir pressure which is given by

$\displaystyle\displaystyle{\cal T}^{33(11)}(x;d)=2\lim_{x^{\prime}\to
x}\sum_{l_{3}=1}^{\infty}{\bigg{\\{}}\left[i\Pi^{33}-\frac{g}{2}\Gamma^{33}\right]G_{0}(x-x^{\prime}-2dl_{3}v_{3}){\bigg{\\}}},$
(68)

where

$\displaystyle\displaystyle\Pi^{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\partial^{0}\partial^{\prime
0}-\partial^{1}\partial^{\prime 1}-\partial^{2}\partial^{\prime
2}+\partial^{3}\partial^{\prime 3}-m^{2},$ (69)
$\displaystyle\displaystyle\Gamma^{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\partial^{\prime 0}\partial^{\prime
0}\partial^{\prime 0}-\partial^{0}\partial^{0}\partial^{0}.$ (70)

After some calculations we find

$\displaystyle\displaystyle{\cal
T}^{33(11)}(d)=-\frac{m^{2}}{4\pi^{2}d^{2}}\sum_{l_{3}=1}^{\infty}\frac{1}{l^{2}_{3}}{\bigg{\\{}}2mdl_{3}K_{1}(2mdl_{3})+3K_{2}(2mdk_{3}){\bigg{\\}}}.$
(71)

Similar to what happens in the Casimir energy, in the Casimir pressure there
is no contribution due to the Lorentz violation when the constant vector
$\displaystyle n^{\mu}$ is a time-like vector. Let us then analyze the cases
in which the constant vector has another direction.

####  IV.2.2 Space-like constant vector

Although there are three possibilities for the case where the vector
$\displaystyle n^{\mu}$ is a space-like constant vector, two of them
$\displaystyle n^{\mu}=(0,1,0,0)$ and $\displaystyle n^{\mu}=(0,0,1,0)$
displays the same result, Eqs. (67) and (71), obtained for the time-like case.
Therefore, special attention should be paid to the case $\displaystyle
n^{\mu}=(0,0,0,1)$.

The last case, which chooses the space-like constant vector as $\displaystyle
n^{\mu}=(0,0,0,1)$, leads to an imaginary Casimir energy. The same happens
with Casimir pressure. However, this application leads to the Casimir effect,
a physical and measurable phenomenon. Therefore, it makes no sense to obtain
an imaginary value for this quantity.

In order to derive real and physical corrections resulting from Lorentz
violation for the Casimir effect, the space-like constant vector must be
chosen as $\displaystyle n^{\mu}=(0,0,0,i)$. For this choice, the components
$\displaystyle\mu=\nu=0$ and $\displaystyle\mu=\nu=3$ of Eq. (46) are given as

$\displaystyle\displaystyle\Gamma^{00}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-i(\partial^{\prime 3}\partial^{\prime
3}\partial^{\prime 3}-\partial^{3}\partial^{3}\partial^{3}),$ (72)
$\displaystyle\displaystyle\Gamma^{33}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-2i(\partial^{3}\partial^{3}\partial^{\prime
3}-\partial^{3}\partial^{\prime 3}\partial^{\prime 3}).$ (73)

With these ingredients the Casimir energy with Lorentz violation is found as

$\displaystyle\displaystyle{\cal T}^{00(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{m^{2}}{4\pi^{2}d^{2}}\sum_{l_{3}=1}^{\infty}\frac{1}{l_{3}^{2}}\Bigl{\\{}K_{2}(2mdl_{3})$
(74) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{g}{2l_{3}}\left[2(dml_{3})^{2}K_{0}(2mdl_{3})+5(dml_{3})K_{1}(2mdl_{3})+6K_{2}(2mdl_{3})\right]\Bigl{\\}}$

and the Casimir pressure is

$\displaystyle\displaystyle{\cal T}^{33(11)}(d)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{m^{2}}{4\pi^{2}d^{2}}\sum_{l_{3}=1}^{\infty}\frac{1}{l_{3}^{2}}\Bigl{\\{}2mdl_{3}K_{1}(2mdl_{3})+3K_{2}(2mdl_{3})$
(75) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{g}{l_{3}}\left[2(dml_{3})^{2}K_{0}(2mdl_{3})+5(dml_{3})K_{1}(2mdl_{3})+6K_{2}(2mdl_{3})\right]\Bigl{\\}}$

It’s important to note that the corrections arising from Lorentz violation
contribute to an increase in both the Casimir energy and pressure.

Looking at the Casimir force, expression (75), in the small mass limit, we get

$\displaystyle\displaystyle{\cal
T}^{00(11)}=-\frac{\pi^{2}}{240d^{4}}-\frac{3g\zeta(5)}{4\pi^{2}d^{5}}.$ (76)

This result demonstrates that the Lorentz violation correction exhibits a
different dependence on $\displaystyle d$, although it remains an attractive
contribution, similar to the usual case.

It is important to observe that there is no contribution to the Stefan-
Boltzmann law due to this case with $\displaystyle n^{\mu}=(0,0,0,i)$.

###  IV.3 Thermal and Size effects

In this subsection, two compactifications are considered, in the time and in
the $\displaystyle z$ direction. As a consequence, thermal and size effects
are analyzed together, i.e., the Casimir effect at finite temperature to the
massive scalar field with higher-order Lorentz symmetry breaking is
calculated. For this application the compactification parameter is assumed as
$\displaystyle\alpha=(\beta,0,0,i2d)$. Then, the Bogoliubov transformation is
written as

$\displaystyle\displaystyle\omega^{2}(\beta,d)=\sum_{l_{0}=1}^{\infty}e^{-\beta
k^{0}l_{0}}+\sum_{l_{3}=1}^{\infty}e^{-i2dk^{3}l_{3}}+2\sum_{l_{0},l_{3}=1}^{\infty}e^{-\beta
k^{0}l_{0}-i2dk^{3}l_{3}},$ (77)

where these terms are associated with the thermal effect, size effect and
thermal and size effects together, respectively. As the interest here is the
combined effects, given by the third term, the Green function associated with
it is given as

$\displaystyle\displaystyle\overline{G}_{0}(x-x^{\prime};\beta,d)=4\sum_{l_{0},l_{3}=1}^{\infty}G_{0}(x-x^{\prime}-i\beta
v_{0}l_{0}-2dl_{3}v_{3}).$ (78)

Now the energy-momentum tensor is calculated for different choices of the
constant vector $\displaystyle n^{\mu}$.

####  IV.3.1 Time-like constant vector

Considering $\displaystyle n^{\mu}=(1,0,0,0)$, the total energy-momentum
tensor with $\displaystyle\mu=\nu=0$ becomes

$\displaystyle\displaystyle{\cal T}^{00(11)}(\beta,d)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle{\cal
T}^{00(11)}_{LI}(\beta,d)+{\cal T}^{00(11)}_{LV}(\beta,d).$ (79)

This give us the Casimir energy at finite temperature with corrections due to
Lorentz violation, where

$\displaystyle\displaystyle{\cal T}^{00(11)}_{LI}(\beta,d)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{2m}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}\Biggl{\\{}\frac{m\left((2dl_{3})^{2}-3(\beta
l_{0})^{2}\right)}{[(2dl_{3})^{2}+(\beta l_{0})^{2}]^{2}}K_{0}\left(z\right)$
(80) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{\left(2(2dl_{3})^{2}-2(\beta
l_{0})^{2}(3+2(dml_{3})^{2})-m^{2}(\beta
l_{0})^{4}\right)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{5/2}}K_{1}\left(z\right)\Biggl{\\}}$

and

$\displaystyle\displaystyle{\cal T}^{00(11)}_{LV}(\beta,d)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{2\beta
gm}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}l_{0}\Biggl{\\{}\frac{m\left((2\beta
l_{0})^{2}((dml_{3})^{2}+3)-12(2dl_{3})^{2}+m^{2}(\beta
l_{0})^{4}\right)}{[(2dl_{3})^{2}+(\beta l_{0})^{2}]^{3}}K_{0}\left(z\right)$
(81) $\displaystyle\displaystyle-$
$\displaystyle\displaystyle\frac{\left(-8(\beta
l_{0})^{2}((dml_{3})^{2}+3)+12(2dl_{3})^{2}((dml_{3})^{2}+2)-5m^{2}(\beta
l_{0})^{4}\right)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{7/2}}K_{1}\left(z\right)\Biggl{\\}}$

with $\displaystyle z=m\sqrt{(2dl_{3})^{2}+(\beta l_{0})^{2}}$. Considering
$\displaystyle\mu=\nu=3$ and following the same steps we get the Casimir
pressure at finite temperature

$\displaystyle\displaystyle{\cal T}^{33(11)}(\beta,d)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle{\cal
T}^{33(11)}_{LI}(\beta,d)+{\cal T}^{33(11)}_{LV}(\beta,d),$ (82)

where

$\displaystyle\displaystyle{\cal T}^{33(11)}_{LI}(\beta,d)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{2m}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}\Biggl{\\{}\frac{m\left(3(2dl_{3})^{2}-(\beta
l_{0})^{2}\right)}{[(2dl_{3})^{2}+(\beta l_{0})^{2}]^{2}}K_{0}\left(z\right)$
(83) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{2\left(-(\beta
l_{0})^{2}+2(dl_{3})^{2}(6+m^{2}(dml_{3})^{2}+(\beta
l_{0})^{2})\right)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{5/2}}K_{1}\left(z\right)\Biggl{\\}}$

and

$\displaystyle\displaystyle{\cal T}^{33(11)}_{LV}(\beta,d)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\frac{\beta
gm}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}l_{0}\Biggl{\\{}\frac{m\left((2\beta
l_{0})^{2}((dml_{3})^{2}+3)-12(2dl_{3})^{2}+m^{2}(\beta
l_{0})^{4}\right)}{[(2dl_{3})^{2}+(\beta l_{0})^{2}]^{3}}K_{0}\left(z\right)$
(84) $\displaystyle\displaystyle-$
$\displaystyle\displaystyle\frac{\left(-8(\beta
l_{0})^{2}((dml_{3})^{2}+3)+12(2dl_{3})^{2}((dml_{3})^{2}+2)-5m^{2}(\beta
l_{0})^{4}\right)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{7/2}}K_{1}\left(z\right)\Biggl{\\}}.$

Therefore, the Lorentz violation changes both the Casimir energy and pressure
at finite temperature. The next step is to investigate this combined effect
considering the space-like constant vector.

####  IV.3.2 Space-like constant vector

Here, we have three possible cases associated with the space-like constant
vector $\displaystyle n^{\mu}$ to analyze. Choosing $\displaystyle
n^{\mu}=(0,1,0,0)$ or $\displaystyle n^{\mu}=(0,0,1,0)$, we find that the
Casimir energy and Casimir pressure at finite temperature are given by Eqs.
(80) and (83), respectively. Then there is no contribution due to Lorentz
violation for this choice of the constant vector $\displaystyle n^{\mu}$.

It is important to note that if $\displaystyle n^{\mu}=(0,0,0,1)$ is chosen,
the contribution associated with the Lorentz violation to the Casimir effect
at finite temperature becomes imaginary. Therefore, since the Casimir effect
is a real phenomenon, the space-like constant vector in the z-direction should
be chosen as $\displaystyle n^{\mu}=(0,0,0,i)$. Then the Lorentz violation
correction for the Casimir energy at finite temperature is given as

$\displaystyle\displaystyle{\cal T}^{00(11)}_{LV}(\beta,d)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{2dgm}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}l_{0}\Biggl{\\{}\frac{4m\left(-(\beta
l_{0})^{2}(3-(dml_{3})^{2})+(2dl_{3})^{2}((mdl_{3})^{2}+3)\right)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{3}}K_{0}\left(z\right)$ (85) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{\left(8(\beta
l_{0})^{2}((dml_{3})^{2}-3)+4(2dl_{3})^{2}(5(dml_{3})^{2}+6)-3m^{2}(\beta
l_{0})^{4}\right)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{7/2}}K_{1}\left(z\right)\Biggl{\\}}$

and for the Casimir pressure at finite temperature is

$\displaystyle\displaystyle{\cal T}^{33(11)}_{LV}(\beta,d)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{4dgm}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}l_{0}\Biggl{\\{}\frac{4m\left(-(\beta
l_{0})^{2}(3-(dml_{3})^{2})+(2dl_{3})^{2}((mdl_{3})^{2}+3)\right)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{3}}K_{0}\left(z\right)$ (86) $\displaystyle\displaystyle+$
$\displaystyle\displaystyle\frac{\left(8(\beta
l_{0})^{2}((dml_{3})^{2}-3)+4(2dl_{3})^{2}(5(dml_{3})^{2}+6)-3m^{2}(\beta
l_{0})^{4}\right)}{[(2dl_{3})^{2}+(\beta
l_{0})^{2}]^{7/2}}K_{1}\left(z\right)\Biggl{\\}}.$

In the very small mass limit, the Casimir energy at finite temperature with
Lorentz violation corrections becomes

$\displaystyle\displaystyle
E(\beta,d)=-\frac{2}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}\left(\frac{(2dl_{3})^{2}-3(\beta
l_{0})^{2}}{[(\beta
l_{0})^{2}+(2dl_{3})^{2}]^{3}}+24gdl_{0}\frac{(2dl_{3})^{2}-(\beta
l_{0})^{2}}{[(\beta l_{0})^{2}+(2dl_{3})^{2}]^{4}}\right)$ (87)

and the Casimir pressure is

$\displaystyle\displaystyle
P(\beta,d)=-\frac{2}{\pi^{2}}\sum_{l_{0},l_{3}=1}^{\infty}\left(\frac{3(2dl_{3})^{2}-(\beta
l_{0})^{2}}{[(\beta
l_{0})^{2}+(2dl_{3})^{2}]^{3}}+48gdl_{0}\frac{(2dl_{3})^{2}-(\beta
l_{0})^{2}}{[(\beta l_{0})^{2}+(2dl_{3})^{2}]^{4}}\right),$ (88)

where $\displaystyle E(\beta,d)\equiv{\cal T}^{00(11)}(\beta,d)$ and
$\displaystyle P(\beta,d)\equiv{\cal T}^{33(11)}(\beta,d)$. Therefore, our
results show that the Lorentz violating extension Myers-Pospelov in the scalar
sector changes the Casimir effect at finite temperature.

##  V Conclusions

Lorentz symmetry is fundamental to the Standard Model (SM) and General
Relativity (GR). However, these theories are not complete. A fundamental
theory must unify both into a single model that describes all interactions.
Theories that seek this unification operate at very high energies, such as the
Planck scale, where tiny violations of Lorentz symmetry emerge. To study these
violations, which lead to new physics, effective theories have been
constructed. One important effective theory developed to investigate Lorentz
and CPT symmetries is the Standard Model Extension (SME), which encompasses
all the physics of the SM and GR, along with additional Lorentz-violating
operators. Here, we consider the scalar sector of the Myers and Pospelov
model. In this model, Lorentz symmetry is broken by the introduction of a
preferred four-vector $\displaystyle n^{\mu}$. The choice of this preferred
four-vector yields different results in the investigation conducted in this
work. To study certain applications arising from the Myers and Pospelov model,
we utilize the TFD formalism. TFD possesses a rich topological structure that
allows the analysis of different effects on an equal footing. Three different
topologies are considered: (1) the topology
$\displaystyle\Gamma_{4}^{1}=\mathbb{S}^{1}\times\mathbb{R}^{3}$, where the
time-axis is compactified. This leads to the Stefan-Boltzmann law. (2) The
topology $\displaystyle\Gamma_{4}^{1}$ with $\displaystyle\alpha=(0,0,0,i2d)$,
where the compactification along the coordinate $\displaystyle z$ is
considered. As a consequence, the Casimir effect at zero temperature is
calculated. (3) The topology
$\displaystyle\Gamma_{4}^{2}=\mathbb{S}^{1}\times\mathbb{S}^{1}\times\mathbb{R}^{2}$
where two axes are compactified, time and coordinate $\displaystyle z$. Then
the Casimir effect at finite temperature is determined. It is important to
note that our results demonstrate the dependence of the corrections to the
Casimir effect and Stefan-Boltzmann law, associated with the massive scalar
field in the presence of the Myers and Pospelov term, on the choice of the
four-vector $\displaystyle n^{\mu}$. Different outcomes arise depending on
whether this four-vector is timelike or spacelike.

## Acknowledgments

This work by A. F. S. is partially supported by National Council for
Scientific and Technological Development - CNPq project No. 312406/2023-1. L.
H. A. R. F. thanks CAPES for financial support. The work of C.M.R has been
partially supported by project Fondecyt Regular 1241369.

## References

  * [1] V. A. Kostelecky and R. Potting, “CPT, strings, and meson factories”, Phys. Rev. D 51, 3923 (1995).
  * [2] V. A. Kostelecky and R. Potting, “CPT and strings”, Nucl. Phys. B 359, 545 (1991).
  * [3] D. Colladay and V. A. Kostelecky, “CPT violation and the standard model”, Phys. Rev. D 55, 6760 (1997).
  * [4] D. Colladay and V. A. Kostelecky, “Lorentz violating extension of the standard model”, Phys. Rev. D 58, 116002 (1998).
  * [5] R. C. Myers and M. Pospelov, “Ultraviolet modifications of dispersion relations in effective field theory”, Phys. Rev. Lett. 90, 211601(2003).
  * [6] L. Balart, C. M. Reyes, S. Ossandon and C. Reyes, “Perturbative unitarity and higher-order Lorentz symmetry breaking”, Phys. Rev. D 98, 035035 (2018).
  * [7] C. M. Reyes, L. Urrutia and J. D. Vergara, “The scalar sector in the Myers-Pospelov model”, AIP Conf. Proc. 977, 214 (2008).
  * [8] Y. Takahashi and H. Umezawa, “Thermo field dynamics”, Int. Jour. Mod. Phys. B 10, 1755 (1996).
  * [9] Y. Takahashi, H. Umezawa and H. Matsumoto, Thermofield Dynamics and Condensed States, North-Holland, Amsterdan, (1982).
  * [10] H. Umezawa, Advanced Field Theory: Micro, Macro and Thermal Physics, AIP, New York, (1993).
  * [11] F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malboiusson and A. E. Santana, Themal quantum field theory: Algebraic aspects and applications, World Scientific, Singapore, (2009).
  * [12] A. E. Santana and F. C. Khanna, “Lie groups and thermal field theory”, Phys. Lett. A 203, 68 (1995).
  * [13] A. E. Santana, F. C. Khanna, H. Chu, and C. Chang, “Thermal Lie Groups, Classical Mechanics, and Thermofield Dynamics”, Ann. Phys. 249, 481 (1996).
  * [14] F. C. Khanna, A. P. C Malbouisson, J. M. C. Malbouisson and A. E. Santana, “Quantum fields in toroidal topology,” Ann. Phys. 326, 2634 (2011).
  * [15] N. Moeller and B. Zwiebach, “Dynamics with infinitely many times derivatives and rolling tachyons”, JHEP 10, 034 (2002).
  * [16] T. C. Cheng, P. M. Ho and M. C. Yeh, “Perturbative approach to higher derivative and nonlocal theories,” Nucl. Phys. B 625, 151-165 (2002), arXiv:hep-th/0111160 [hep-th]. T. C. Cheng, P. M. Ho and M. C. Yeh, “Perturbative approach to higher derivative theories with fermions,” Phys. Rev. D 66, 085015 (2002), arXiv:hep-th/0206077 [hep-th].