footnote

11institutetext:  Southern University of Science and Technology
22institutetext: Swinburne University of Technology
33institutetext: University of Birmingham
44institutetext: CSIRO Data61

# Non-Fungible Token (NFT): Overview, Evaluation, Opportunities and Challenges

(Tech Report${}^{\textrm{V}2}$)

Qin Wang⋆ 2244 Rujia Li These authors contributed equally to the work.
Email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Qi Wang 11 Shiping Chen
44

###### Abstract

The Non-Fungible Token (NFT) market is mushrooming in recent years. The
concept of NFT originally comes from a token standard of Ethereum, aiming to
distinguish each token with distinguishable signs. This type of token can be
bound with virtual/digital properties as their unique identifications. With
NFTs, all marked properties can be freely traded with customized values
according to their ages, rarity, liquidity, etc. It has greatly stimulated the
prosperity of the decentralized application (DApp) market. At the time of
writing (May 2021), the total money used on completed NFT sales has reached
$34,530,649.86$ USD. The thousandfold return on its increasing market draws
huge attention worldwide. However, the development of the NFT ecosystem is
still in its early stage, and the technologies of NFTs are pre-mature.
Newcomers may get lost in their frenetic evolution due to the lack of
systematic summaries. In this technical report, we explore the NFT ecosystems
in several aspects. We start with an overview of state-of-the-art NFT
solutions, then provide their technical components, protocols, standards, and
desired proprieties. Afterwards, we give a security evolution, with
discussions on the perspectives of their design models, opportunities and
challenges. To the best of our knowledge, this is the first systematic study
on the current NFT ecosystems.

###### Keywords:

Blockchain NFT DApp Smart contract

##  1 Introduction

Non-Fungible Token (NFT) is a type of cryptocurrency [82] that is derived by
the smart contracts of Ethereum [119]. NFT was firstly proposed in Ethereum
Improvement Proposals (EIP)-721 [116] and further developed in EIP-1155 [117].
NFT differs from classical cryptocurrencies [105] such as Bitcoin [98] in
their intrinsic features. Bitcoin [98] is a standard coin in which all the
coins are equivalent and indistinguishable. In contrast, NFT is unique which
cannot be exchanged like-for-like (equivalently, non-fungible), making it
suitable for identifying something or someone in a unique way. To be specific,
by using NFTs on smart contracts (in Ethereum [119]), a creator can easily
prove the existence and ownership of digital assets in the form of videos,
images, arts [83], event tickets [103], etc. Furthermore, the creator can also
earn royalties each time of a successful trade on any NFT market or by peer-
to-peer exchanging. Full-history tradability, deep liquidity, and convenient
interoperability enable NFT to become a promising intellectual property
(IP)-protection solution. Although, in essence, NFTs represent little more
than code, but the codes to a buyer have ascribed value when considering its
comparative scarcity as a digital object. It well secures selling prices of
these IP-related products that may have seemed unthinkable for non-fungible
virtual assets.

In recent years, NFTs have garnered remarkable attention from both the
industrial and scientific communities. It was reported that the 24-hour
trading volume on average of the NFT market is $4,592,146,914$ USD111Data
source from Coingecko (May, 2021) https://www.coingecko.com/en/nft, while the
24-hour trading volume of the entire cryptocurrency market is
$341,017,001,809$ USD. The liquidity of NFT-related solutions has accounted
for 1.3$\%$ of the entire cryptocurrency market in such a short period (5
months). Early investors obtain thousandfold returns by selling unique digital
collectibles. At the time of writing (May 2021), the NFTs-related market has
significantly increased compared to one year ago (January 2020). Specifically,
the total number of sales is $25,729$ and their total amounts spent on
completed sales reach $34,530,649.86$ USD222Data captured from
https://nonfungible.com/market/history. In particular, the total number of
primary-market sales occupies $17,140$, while the number of secondary sales
(user-to-user) is $8,589$. Correspondingly, the total USD used on primary
market sales is $8,816,531.10$. Besides, the active market wallets achieve
$12,836$, which is still increasing at a high speed as time goes.
Surprisingly, the sale of NFTs was estimated at $12$ million (December 2020)
but exploded to $340$ million within just two months (February 2021). Such
skyrocketing development makes NFT become a craze, or even be described by
some as the future of digital assets.

Besides the above data, people have expressed interest in various types of
NFTs. They participate in NFT-related games or trades with enthusiasm.
CryptoPunks [20], one of the first NFT on Ethereum, has created more than
$10,000$ collectible punks ($6039$ males and $3840$ females) and further
promoted the ERC-721 standard to become popular. CryptoKitties [19] officially
put NFTs on notice, and hit the market in 2017 with the gamification of the
breeding mechanics. Participants fiercely competed at high prices to auction
the rare cats, and the highest price reaches more than 999 ETH333Price
available: https://www.cryptokitties.co/kitty/1866420 (equally 3M USD).
Another outstanding instance is NBA Top Shot [44], which is an NFT trading
platform used to buy/sell digital short videos of NBA moments. Thousands of
NBA fans from around worldwide have collected over $7.6$ million top shot
moments, building the roster of rookies, vets, and rising star players.
Following projects also enjoy great success including Picasso Punks [51],
Hashmasks [31], 3DPunks [4], unofficial punks [63], Polkamon [52], Chubbies
[14], Bullrun Babes [12], Aavegotchi [5], CryptoCats [18], Moon Cats Rescue
[42], NFT box [46], etc. There is no doubt that there is a hype cycle
surrounding NFTs where most of products can be sold with high prices, some
even hundreds or thousands of ETHs. Besides games and collectibles, NFTs also
promote the development of art [108], ticketing event [103], value [77],[78],
IoT [100], and finance [80][97]. Other types of surrounding markets play
important roles as well to provide instant information and secure environments
like statistic websites (e.g. NonFungible [49], DappRadar [24], NFT bank[45],
DefiPulse [26], Coingecko [15]), trading marketplace (cryptoslam [21], Opensea
[50], SuperRare [59], Nifty Gateway [48], Rarible [55], Zora [69]) and so-
called NFT ecosystem (such as Dego [25]).

Despite NFTs have a tremendous potential impact on the current decentralized
markets and future business opportunities, the NFT technologies are still in
the very early stage. Some potential challenges are required to be carefully
tackled, while some promising opportunities should be highlighted. Further,
even though much literature on NFTs, from blogs, wikis, forum posts, codes and
other sources, are available to the public, a systematic study is absent. This
paper aims to draw attention to these questions insofar as observed and focus
on summarising current NFT solutions. We provide a detailed analysis of its
core components, technology roadmap status, opportunities and challenges. The
contributions are provided as follows.

  * -

Firstly, we abstract the design models of current NFT solutions. Specifically,
we identify the core technical components that are used to construct NFTs.
Then, we present their protocols, standards and targeted properties.

  * -

Secondly, we give a security evaluation of current NFT systems. We adopt the
STRIDE threat and risk evaluation [106] to investigate potential security
issues. Based on that, we also discuss the corresponding defense measures for
the issues.

  * -

Thirdly, we explore some future opportunities of NFTs in many fields. Applying
NFTs to real scenarios will boost a wide range of new applications. We give a
set of practical instances (projects) leveraging NFTs with great success or
prosperous markets.

  * -

Finally, we highlight a series of open challenges in NFT ecosystems.
Blockchain-based NFT systems still confront unavoidable problems like privacy
issues, data inaccessibility, etc. We outline open challenges existed in
state-of-the-art NFT solutions.

The rest part of this work is organized as follows. Section 2 provides the
technical components used to build NFTs. Section 3 presents the protocols and
standards. Based on that, Section 4 gives our security evaluation. Section 5
discusses the future opportunities, while Section 6 outlines the open
challenges. Finally, Section 7 concludes this work. Appendix A gives our
version updates. Appendix B and Appendix C provide NFT collectible ranking and
an overview of existing NFT projects, respectively. Appendix D presents a
detailed instance analysis.

##  2 Technical Components

In this part, we show technical components related to the NFT’s activities.
These components lay the building foundations of a fully functional NFT
scheme.

Blockchain. Blockchain was originally proposed by Nakamoto [98], where Bitcoin
uses the proof of work (PoW) [87] algorithm to reach an agreement on
transaction data in a decentralized network. Blockchain is defined as a
distributed and attached-only database that maintains a list of data records
linked and protected using cryptographic protocols [86]. Blockchain provides a
solution to the long-standing Byzantine problem [92], which has been agreed
upon with a large network of untrusted participants. Once the shared data on
the blockchain is confirmed in most distributed nodes, it becomes immutable
because any changes in the stored data will invalidate all subsequent data.
The most prevailing blockchain platform used in NFT schemes is Ethereum [119],
providing a secure environment for executing the smart contracts. In addition,
several solutions drop their customized chain-engines or blockchain platforms
to support their specialized applications, and some of them are Flow [2], EOS
[67], Hyperledger [89][70], and Fast Box [28][111].

Smart Contract. Smart contracts were originally introduced by Szabo [107],
aiming to accelerate, verify or execute digital negotiation. Ethereum [119]
further developed smart contracts in the blockchain system [84][71].
Blockchain-based smart contracts adopt Turing-complete scripting languages to
achieve complicated functionalities and execute thorough state transition
replication over consensus algorithms to realize final consistency. Smart
contracts enable unfamiliar parties and decentralized participants to conduct
fair exchanges without a trusted third party and further propose a unified
method to build applications across a wide range of industries. The
applications operating on top of smart contracts are based on state-transition
mechanisms. The states that contain the instructions and parameters are shared
by all the participants, thus guaranteeing transparency of the execution of
these instructions. Also, the positions between states have to stay the same
across distributed nodes, which is important to its consistency. Most NFT
solutions [20][19][44][51][31][4] rely on smart contract-based blockchain
platforms to ensure their order-sensitive executions.

Address and Transaction. Blockchain address and transaction are the essential
concepts in cryptocurrencies. A blockchain address is a unique identifier for
a user to send and receive the assets, which is similar to a bank account when
spending the assets in the bank. It consists of a fixed number of alphanumeric
characters generated from a pair of public key and private key. To transfer
NFTs, the owner must prove in possession of the corresponding private key and
send the assets to another address(es) with a correct digital signature. This
simple operation is usually performed using a cryptocurrency wallet and is
represented as sending a transaction to involve smart contracts in the ERC-777
[90] standard.

Data Encoding. Encoding is the process of converting data from one form to
another. Normally, many files are often encoded into either efficient,
compressed formats for saving disk space or into an uncompressed format for
high quality/resolution. In the mainstream blockchain systems such as Bitcoin
[98] and Ethereum [119], they employ hex values to encode transaction elements
such as the function names, parameters and return values. This implies that
the raw NFT data must follow these rules. If one claims s/he owns the NFT-
based intellectual property, s/he essentially owns the original piece of hex
values signed by the creator. Others can freely copy the raw data, but they
cannot claim ownership of the property. Based on that, we can observe that the
NFT-related activities (e.g. buy/sell/trade/auction) have to be processed
under these four phases, similar to the basic processing procedure of smart
contracts.

##  3 Protocols, Standards and Properties

This section presents two basic models of NFT schemes, with emphasis on their
protocols, token standards and key properties.

###  3.1 Protocols

The establishment of NFT requires an underlying distributed ledger for
records, together with exchangeable transactions for trading in the peer-to-
peer network. This report primarily treats the distributed ledger as a special
type of database to store NFT data. In particular, we assume that the ledger
has basic security consistency, completeness, and availability
characteristics. Based on that, we identify two design patterns for the NFT
paradigm. The former protocol is established from top to bottom with a very
simple yet classical path: building NFTs from the initiator, and then sell
them to the buyer. In contrast, the later route (e.g. Loot [34], detailed
analysis in Appendix D) reverses this path: setting a NFT template, and every
user can create their unique ontop NFTs. We separately provide detailed
protocols of these two design patterns as below. To be noted, for both of
them, they still follow a very similar workflow when executed on blockchain
systems (cf. Fig.1), meaning that different designs will not change the
underlying operating mechanism.

Figure 1: Workflow of NFT Systems

Top to Bottom. For the first design (e.g., CryptoPunks [20]), an NFT protocol
consists of another two roles: NFT owner and NFT buyer.

  * -

NFT Digitize. An NFT owner checks that the file, title, description are
completely accurate. Then, s/he digitizes the raw data into a proper format.

  * -

NFT Store. An NFT owner stores the raw data into an external database outside
the blockchain. Note that, s/he is also allowed to store the raw data inside a
blockchain, despite this operation is gas-consuming.

  * -

NFT Sign. The NFT owner signs a transaction, including the hash of NFT data,
and then sends the transaction to a smart contract.

  * -

NFT Mint&Trade. After the smart contract receives the transaction with the NFT
data, the minting and trading process begins. The main mechanism behind NFTs
is the logic of the Token Standards that can be found in Section 3.2.

  * -

NFT Confirm. Once the transaction is confirmed, the minting process completes.
By this approach, NFTs will forever link to a unique blockchain address as
their persistence evidence.

Bottom to Top. For this design (e.g., Loot [34]), the protocol consists of two
roles: NFT creator and NFT buyer. In most cases, a buyer can also act as a
creator because an NFT product is created based on random seeds when a buyer
bids for it. This extends the functions in terms of user customization. Here,
we use the superscript $*$ to highlight differences compared with the previous
one.

  * -

Template Create∗. The project founder initiates a template via the smart
contract to set up several basic rules, such as different features (character
style, weapons, or accessories) in the game.

  * -

NFT Randomize∗. Once a buyer bids for an NFT, s/he can customize the NFT
product with a set of additional features on top of basic lines. These
additional features are randomly selected from a database that was predefined
at the initial state.

  * -

NFT Mint&Trade. The minting and trading process starts once the corresponding
smart contract is triggered.

  * -

NFT Confirm. All the procedures are conducted through smart contracts. The
generated NFT will be persistently stored on-chain when the consensus
procedure has been completed.

In a blockchain system, each block has a limited capacity. When the capacity
in one block becomes full, other transactions will enter a future block linked
to the original data block. In the end, all linked blocks have created a long-
term history that remains permanent. The NFT system, in essence, is a
blockchain-based application. Whenever an NFT is minted or sold, a new
transaction is required to send to invoke the smart contract. After the
transaction is confirmed, the NFT metadata and ownership details are added to
a new block, thereby ensuring that the history of the NFT remains unchanged
and the ownership is preserved.

###  3.2 Token Standards

In this part, we clarify token standards related to NFTs, including ERC-20
[81], ERC-721 [115], and ERC-1155 [118] (see Algorithm 1). These standards
have a great impact on the ongoing NFT schemes. We discuss them as follows.

interface ERC721 {

function ownerOf(uint256$\\_$tokenId) external view returns (address);

function transferFrom(address$\\_$from, address$\\_$to, uint256$\\_$tokenId)
external payable; …

}

interface ERC1155 {

function balanceOf(address$\\_$owner, uint256$\\_$id) external view returns
(address);

function balanceOfBatch(address calldata $\\_$owners, uint256 calldata
$\\_$ids) external view returns (uint256 memory);

function transferFrom(address$\\_$from, address$\\_$to, uint256$\\_$id,
uint256 quantity) external payable; …

}

Algorithm 1 NFT Standard Interfaces (with selected functions)

The most prevailing token standard comes from ERC-20 [81]. It introduces the
concept of fungible tokens that can be issued on top of Ethereum once
satisfying the requirements. The standard makes tokens the same as another one
(in terms of both type and value). An arbitrary token is always equal to all
the other tokens. This stimulates the hype of Initial Coin Offering (ICO) from
2015 to present. A lot of public chains and various blockchain-based DApps
[75][102] gain sufficient initial fundings in this way. In contrast, ERC-721
[115] introduces a non-fungible token standard that differs from the fungible
token. This type of token is unique that can be distinguished from another
token. Specifically, every NFT has a uint256 variable called tokenId, and the
pair of contract address and uint256 tokenId is globally unique. Further, the
tokenId can be used as an input to generate special identifications such as
images in the form of zombies or cartoon characters.

Another standard ERC-1155 (Multi Token Standard) [118] extends the
representation of both fungible and non-fungible tokens. It provides an
interface that can represent any number of tokens. In previous standards,
every tokenId in contact only contains a single type of tokens. For instance,
ERC-20 makes each token type deployed in separate contracts. As well, ERC-721
deploys the group of non-fungible tokens in a single contract with the same
configurations. In contrast, ERC-1155 extends the functionality of tokenId,
where each of them can independently represent different configurable token
types. The field may contain its customized information such as the metadata,
lock-time, date, supply, or any other attributes. Here, we provide an
illustration (see Fig.2) to show their structures and aforementioned
differences.

Figure 2: NFT-related Token Standards

###  3.3 NFTs Desired Proprieties

NFT schemes are essentially decentralized applications [74], and thus enjoy
the benefits/properties from their underlying public ledgers. We summarise the
key properties as follows.

  * -

Verifiability. The NFT with its token metadata and its ownership can be
publicly verified.

  * -

Transparent Execution. The activities of NFTs include minting, selling and
purchasing are publicly accessible.

  * -

Availability. The NFT system never goes down. Alternatively, all the tokens
and issued NFTs are always available to sell and buy.

  * -

Tamper-resistance. The NFT metadata and its trading records are persistently
stored and cannot be manipulated once the transactions are deemed as
confirmed.

  * -

Usability. Every NFT has the most up-to-date ownership information, which is
user-friendly and information-clearly.

  * -

Atomicity. Trading NFTs can be completed in one atomic, consistent, isolated,
and durable (ACID) transaction. The NFTs can run in the same shared execution
state.

  * -

Tradability. Every NFTs and its corresponding products can be arbitrarily
traded and exchanged.

##  4 Security Evaluation

An NFT system is a combination technology that consists of blockchain, storage
and web application. Security evaluation on the NFT system is challenging
since each component may become an attacking interface that makes the whole
system really vulnerable against the attacker. Thus, we adopt the STRIDE
threat and risk evaluation [106], which covers all security aspects of a
system: authenticity, integrity, non-repudiability, availability and access
control. We investigate the potential security issues and propose some of the
corresponding defense measures to address these issues (see Table 1).

Spoofing. Spoofing is the ability to impersonate another entity (for example,
another person or computer) on the system, which corresponds to authenticity.
When a user interacts to mint or sells NFTs, a malicious attacker may exploit
authentication vulnerabilities or steal the user’s private key to transfer the
ownership of NFTs illegally. Thus, we recommend having a formal verification
for the NFT smart contract and to use the cold wallet to prevent private key
leakage.

Tampering. Tampering refers to the malicious modification of NFT data, which
violates integrity. Assume that the blockchain is a robust public transaction
ledger [85, 86] and a hash algorithm is preimage resistance and second
preimage resistance [104]. The metadata and ownership of NFTs cannot be
maliciously modified after the transaction is confirmed. However, the data
stored outside blockchain may be manipulated. Therefore, we recommend users to
send both the hash data as well as the original data to the NFT buyer when
trading/exchanging NFT-related properties.

Repudiation. Repudiation refers to the situation where the author of a
statement cannot dispute [121], which is related to the security property of
non-repudiability [95]. In particular, the fact that a user sends NFT to
another user cannot deny. This is guaranteed by the security of the blockchain
and the unforgeability property of a signature scheme. However, the hash data
may be tampered by a malicious attacker, or the hash data may bind with an
attacker’s address. Thus, we believe that using a multi-signature contract can
partly solve this issue since each binding must be confirmed by more than one
participant.

Information Disclosure. Information leakage occurs when information is exposed
to unauthorized users, which violates confidentiality [95]. In the NFT system,
the state information and the instruction code in the smart contracts are
entirely transparent, and any state and its changes are publicly accessible by
any observer. Even if the user only puts the NFT hash into the blockchain, the
malicious attackers can easily exploit the linkability of the hash and
transaction. Thus, we recommend the NFT developer to use privacy-preserving
smart contracts [93][94] instead of plain smart contracts to protect the
user’s privacy.

=0pt =80pt  STRIDE Security Issues Solutions Spoofing (Authenticity) • An
attacker may exploit authentication vulnerabilities • An attacker may steal a
user’s private key. • A formal verification on the smart contract. • Using the
cold wallet to prevent the private key leakage. Tampering (Integrity) • The
data stored outside the blockchain may be manipulated. • Sending both the
original data and hash data to the NFT buyer when trading NFTs. Repudiation
(Non-repudiability) • The hash data may bind with an attacker’s address. •
Using a multi-signature contract partly. Information disclosure
(Confidentiality) • An attacker can easily exploit the hash and transaction to
link a particular NFT buyer or seller. • Using privacy-preserving smart
contracts instead of smart contracts to protect the user’s privacy. Denial of
service (Availability) • The NFT data may become unavailable if the asset is
stored outside the blockchain. • Using the hybrid blockchain architecture with
weak consensus algorithm. Elevation of privilege (Authorization) • A poorly
designed smart contract may make NFTs lose such properties. • A formal
verification on the smart contracts.

Table 1: Potential Security Issues and Corresponding Solutions of NFTs.

Denial of Service (DoS). DoS attack [96] is a type of network attack in which
a malicious attacker aims to render a server unavailable to its intended users
by interrupting the normal functions. DoS violates the availability and breaks
down the NFT service, which can indeed be used by unauthorized users.
Fortunately, the blockchain guarantees the high availability of user’s
operations. Legitimate users can use the required information when needed and
will not lose data resources due to accidental errors. However, DoS can also
be used to attack the centralized web applications or the raw data outside the
blockchain, resulting in denial-of-service to NFT service. Recently, a new
hybrid blockchain architecture with weak consensus algorithm was proposed
[111], by which this architecture solves the availability issues using two
algorithms.

Elevation of Privilege. Elevation of Privilege [106] is a property that is
related to the authorization. In this type of threat, an attacker may gain
permissions beyond those initially granted. In the NFT system, the selling
permissions are managed by a smart contract. Again, a poorly designed smart
contract may make NFTs lose such properties.

##  5 Opportunities

This section explores the opportunities of NFTs. We discuss several typical
fields which may get benefits from NFTs.

Boosting Gaming Industry. NFT has great potential in the gaming industry.
There already exist some crypto games are CrytpoKitties [19], Cryptocats [18],
CryptoPunks [20], Meebits [38], Axie Infinity [9], Gods Unchanged [29], and
TradeStars [61]. A fascinating feature of such games is the “breeding”
mechanism. Users can personally raise pets and spend much time breeding new
offspring. They can also purchase the limited/rare edition virtual pets, and
then sell them at a high price. The extra reward attracts lots of investors to
join the games, making NFTs come to prominence. Another exciting functions of
the NFT are that it provides ownership records of items in the games and
promotes economic marking place in the ecosystem, benefiting both developers
and players. In particular, game developers who are NFT publishers of the
features (e.g., weapons and skins) can earn royalties each time their items
are (re-)sold on the open market. The players can obtain personal exclusivity
game items. This will create a mutually beneficial business model in which
both players and developers profit from the secondary NFT market. After that,
blockchain communities extend NFTs to a large extent that covers various types
of digital assets.

Flourishing Virtual Events. Traditional online events rely on centralized
companies that provide trust and technology. Although blockchain takes over
several types of activities like raising money (either by ICO/IFO/IEO/etc.),
its applications are still constrained in a small range of events. NFTs
greatly extend the scope of blockchain applications with the help of their
additional properties (uniqueness, ownership, liquidity). This enables each
individual to link to a specific event just like the patterns in our real
life. We give the instance of the ticketing event. When buying tickets in a
traditional event ticket market, consumers must trust the third party.
Therefore, there is a risk of buying fraudulent or invalid tickets, which are
possibly counterfeit or might be cancelled. The same ticket may be sold many
times or obtained by extracting from ticket images posted online in an extreme
case. “NFT-based ticket” represents a ticket issued by the blockchain to
demonstrate entitlement to access to any event such as culture or sports. An
NFT-based ticket is unique and scarce, meaning that the ticket holder cannot
resell the ticket after it is sold. The blockchain-based smart contract
provides a transparent ticket trading platform for the stakeholders such as
the event organizer and the customer. Consumers can buy and sell the crypto
ticket from the smart contract rather than rely on third parties in an
efficient and reliable way.

Protecting Digital Collectibles. Digital collectibles contain a variety of
types, ranging from trading cards, wines [68], digital images [59], videos
[44], virtual real estate [1], domain names [27][64], diamonds [32][47],
crypto stamps [17] and other real/intellectual properties. We take the field
of arts as an example. Firstly, artists in traditional ways have very few
channels to display the works. The prices cannot reflect the true value of
their works due to the absence of attention. Even worse, their published work
on social networks has been charged with intermediary fees by platforms and
advertisements. NFTs transform their work into digital formats with integrated
identities. Artists do not have to transfer ownership and contents to agents.
This provides them impetus with lots of profits. Typical instances include Mad
Dog Jones’s REPLICATOR (selling with 4.1 million USD [36]), Grimes’s work
(selling in total around 6 million USD [30]) and other works from great
crypto-artists like Beeple [10] / Trevor Jones [62]. Furthermore, artists in
general cases cannot receive royalties from future sales of their works. In
contrast, NFTs can be programmed so that the artist receives a predetermined
royalty fee each time when his digital artwork exchanges in the markets (e.g.
[33], SuperRare [59], MakersPlace [37], Rare Art Lab[54], VIV3 [65]). This is
an efficient way to manage and protect digital masterpieces. In addition,
several platforms (e.g. Mintbase [41], Mintable [40]) have even established
tools to support ordinary people to create their own NFT works easily.

Inspiring the Metaverse. Metaverse is a collective virtual shared space that
allows all types of digital activities. Generally, it covers a set of
techniques like augmented reality and the Internet to establish the virtual
world. The concept stems from the last decades and has a great progress with
the rapid development of blockchain. Blockchain provides an ideal
decentralized environment for the virtual online world. Participants under
this blockchain-fueled alternative realities [50] can have many types of
intriguing use cases like enjoying games, displaying self-made arts, trading
assets and virtual properties (arts, land parcels, names, video shots,
wearables), etc. In addition, users also have opportunities to get profits
from the virtual economy. They can lease the buildings (such as offices) to
others to earn the bond or raise rare pets and sell them to get the rewards.
Primary blockchain-empowered projects are Decentraland [1], Cryptovoxels [22],
Somnium Space [57], MegaCryptoPolis [39] and Sandbox [3]. In fact, the
metaverse ecosystem covers all aforementioned applications. We list it
separately here simply because it is still in an early stage due to the
complexity.

##  6 Challenges

To enable the development of the above NFT applications, a series of barriers
have to be overcome as with any nascent technologies. We discuss some typical
challenges from the perspectives of usability, security, governance, and
extensibility, covering both the system level issues caused by blockchain-
based platforms and human factors such as governs, regulation, and society.

###  6.1 Usability Challenges

Usability is to measure the users’ effectiveness, efficiency, and satisfaction
when testing a specific product/design. Most NFT schemes are built on top of
Ethereum. Therefore, it is obvious that the main drawbacks of Ethereum still
exist. We discuss two major challenges that have direct impacts on the user
experience.

  * -

Slow Confirmation. NFT-related procedures are tyically conducted by sending
transactions via the smart contract for reliable and transparent management
(such as mint, sell, exchange). However, current NFT systems are closely
coupled with their underlying blockchain platforms, which makes them suffer
from low performance (Bitcoin reaches merely 7 TPS [109] while Ethereum only
30 TPS). This results in extremely slow confirmation of NFTs. Conquering this
issue requires a redesign of blockchain systems [113], optimization of its
structure [110][88] or improvement on the consensus mechanisms [71]. Existing
blockchain systems cannot fulfil such requirements.

  * -

High Gas Prices. High gas prices have become a major problem for NFT
marketplaces, especially when minting the NFTs at a large scale that requires
uploading the metadata to the blockchain network. Every NFT-related
transactions are more expensive than a simple transfer transaction because
smart contracts involve computational resources and storage to be processed.
At the time of writing, to mine an NFT token costs over USD $60$ (equivalently
in around $5\times 10^{2}wei$) 444Source calculated from
https://coinmarketcap.com/ and https://ethereum.org/en/developers/docs/gas/.
To complete a simple NFT trade can run between USD $60$ and USD $100$ for each
transaction. Expensive fees caused by complex operations and high congestion
greatly limit its wide adoption.

###  6.2 Security and Privacy Issues

The security of user data pose the first priority of systems. However, the
data (stored off-chain but relates to on-chain tags) confronts the risk of
losing linkage or being misused by malicious parties. We give details as
follows.

  * -

NFT Data Inaccessibility. In the mainstream NFT projects [2, 1, 3], a
cryptographic “hash” as the identifier, instead of a copy of the file, will be
tagged with the token and then recorded on the blockchain to save the gas
consumption. This makes the user lose confidence in the NFT because the
original file might be lost or damaged. Several NFT projects integrate their
system with a specialized file storage system such as IPFS [72] in which IPFS
addresses allow users to find a piece of content so long as someone somewhere
on the IPFS network is hosting it. Inevitably, such systems have flaws. When
the users “upload” NFT metadata to IPFS nodes, there is no guarantee that
their data will be replicated among all the nodes. The data may become
unavailable if the asset is stored on IPFS and the only node storing it is
disconnected from the network [73]. This issue has been reported by DECRYPT.IO
[73] and CHECKMYNFT.COM [76]. Also, an NFT might point to an erroneous file
address. If that is the case, a user cannot prove that s/he actually owns the
NFT. In a word, relying on an external system as the core component (storage)
for an NFT system is vulnerable.

  * -

Anonymity/Privacy. In the current stage, the anonymity and privacy of NFTs are
still understudied. Most NFT transactions rely on their underlying Ethereum
platform, which only provides pseudo-anonymity rather than strict anonymity or
privacy. Users can partially hide their identities if the links between their
real identities and corresponding addresses are unknown by the public.
Otherwise, all the activities of users under the exposed address are
observable. Existing privacy-preserving solutions (e.g. homomorphic encryption
[112], zero-knowledge proof [114], ring signature [99], multi-party
computation [101]) have not been yet applied to the NFT-related schemes due to
their complicated cryptographic primitives and security assumptions. Similar
to other types of blockchain-based systems, decreasing expensive computation
costs becomes the key to implement privacy-promised schemes.

###  6.3 Governance Consideration

Similar to the situations of most cryptocurrencies, NFTs also confront the
barriers like strict management from the government. On the other side, how to
properly regulate this nascent technology with the corresponding market is
also a challenge. We discuss two typical issues from both sides.

  * -

Legal Pitfalls. NFTs confront legal and policy issues across a wide range of
areas [82][91]. Potential concerned areas cover commodities, cross-border
transactions, KYC (Know Your Customer) data, etc. It is important to
understand the related regulatory scrutiny and litigation before moving into
the NFT tracks. In some countries, such as Indian and China, the legal
situation is strict for cryptocurrencies, and also for NFT sales. Exchanging,
trading, selling, or buying NFTs have to overcome the difficulties of
governance. Legally, users can only trade derivates on authorized exchanges
such as stocks and commodities or exchange tokens with someone person-to-
person. Several countries, such as Malta and France, are trying to implement
suitable laws with the aim to regulate the service of digital assets.
Elsewhere, issues are resolved by using existing laws. They require buyers to
follow complex or even contradictory terms. Therefore, undertaking due
diligence is a necessity before investing serious tokens in NFTs.

  * -

Taxable property Issues. IP-related products (including arts, books, domain
names, etc.) are treated as taxable property under the current legal
framework. However, NFT-based sales stay out of this scope. Although few
countries, such as the U.S. (internal revenue service, IRS), tax
cryptocurrencies as property, most areas worldwide have not yet considered it.
This may greatly increase the financial crimes under cover of NFT trading. The
governments would love to make the sale of NFTs reliable with tax
consequences. Specifically, the individual participants should have the tax
liability on any capital gains that are related to NFT properties. Also, NFT-
for-NFT, NFT-for-IP, and Eth-for-NFT (or vice versa) exchanges should be
taxed. Furthermore, for high-profit properties, or collectibles, a higher tax
bracket should be applied. Thus, NFT-related trades are suggested to seek more
advice from professional tax departments after the profound discussions.

###  6.4 Extensibility Issues

The extensibility of NFT schemes is two-fold. The first is to stress whether a
system can interact with other ecosystems. The second focuses on whether NFT
systems can obtain updates when the current version is left behind.

  * -

NFT Interoperability (cross-chain). Existing NFT ecosystems are isolated from
each other. Users once have selected one type of product can only
sell/buy/trade them within the same ecosystem/network. This is due to the
reason of its underlying blockchain platform. Interoperability and cross-chain
communication are always the handicaps for the wide adoption of DApps. Based
on the observations from [120], cross-chain communications can only be
implemented with the help of external trusted parties. The decentralization
property, in this way, has been inevitably lost to some extent. But
fortunately, most of the NFT-related projects adopt Ethereum as their
underlying platform. This indicates that they share a similar data structure
and can exchange under the same rules.

  * -

Updatable NFTs. Transitional blockchains update their protocols through soft
forks (minor modifications that are compatible forwards) and hard forks
(significant modifications that may conflict with previous protocols). A
formal discussion has been provided in [79] stating the difficulties and
trade-offs when applying the updates to an existing blockchain. Despite using
the generic model, a new version still faces strict requirements such as
tolerating specific adversarial behaviours and staying online during the
update process. NFT schemes closely rely on their underlying platforms and
keep consistent with them. Although the data are often stored in separate
components (such as the IPFS file system), the most important logic and tokeId
are still recorded on-chain. Properly updating the system with improvements
will be a necessity.

##  7 Conclusion

Non-Fungible Token (NFT) is an emerging technology prevailing in the
blockchain market. In this report, we explore the state-of-the-art NFT
solutions which may re-shape the market of digital/virtual assets stepping
forward. We firstly analyze the technical components and provide the design
models and properties. Then, we evaluate the security of current NFTs systems
and further discuss the opportunities and potential applications that adopt
the NFT concept. Finally, we outline existing research challenges that require
to be solved before achieving mass-market penetration. We hope this report
delivers timely analysis and summary of existing proposed solutions and
projects, making it easier for newcomers to keep up with the current progress.

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## Appendix A. Version Control

This work is an ongoing updated technique report. In the future, we will add
the landmark events from in the wild NFT projects to inspire technique
movements and developments. Dynamic data (like prices, sales, or market cap)
will remain as it is in the current snapshot [as of May 2021].

Table 2: Version Updates

Version | Date | Updates
---|---|---
V1 | May 2021 |  Main body construction
V2 | October 2021 |  Adding the bottom to top model in Section 3
|  |  Adding the instance analysis of the Loot project in Appendix D
|  |  Revisions of unclear sentences and paragraphs

## Appendix B. Top Sales of NFT Properties

Table 3:  NFT Collectible Ranking by Sales Volume (All-time)

Rank | Product |  Sales |  Buyers |  Txns |  Owners
---|---|---|---|---|---
1 | NBA Top Shot [44] |  $\$$573,815,060.38 |  285,306 |  4,877,975 |  498,659
2 | CryptoPunks [20] |  $\$$316,956,115.15 |  2,716 |  13,231 |  2,282
3 | Meebits [38] |  $\$$54,750,807.84 |  1,336 |  2,995 |  4,436
4 | Hashmasks [31] |  $\$$49,713,202.76 |  3,165 |  11,322 |  4,254
5 | Sorare [58] |  $\$$39,807,257.19 |  16,435 |  225,909 |  18,050
6 | CryptoKitties [19] |  $\$$33,230,422.66 |  100,624 |  762,562 |
7 | Art Blocks [7] |  $\$$18,824,119.18 |  2,052 |  10,907 |  4,521
8 | Alien Worlds [6] |  $\$$17,239,575.21 |  165,818 |  2,280,968 |  1,124,901
9 | Bored Ape Yacht Club [11] |  $\$$11,844,235.71 |  2,281 |  5,786 |  2,697
10 | Topps MLB [60] |  $\$$10,205,104.07 |  12,706 |  365,963 |  36,283

  * 1

Data captured on May 10, 20201, source from: https://cryptoslam.io/

## Appendix C. Overview of Existing NFT solutions

Table 4: Existing NFTs Projects (May, 2021)

Types | Projects | Platform |  Key Words
---|---|---|---
Collectible | NBA Top Shot [44] | Flow [2] | Basketball moments
CryptoPunks [20] | Ethereum | The first NFT products
CryptoWine [23] | Ethereum | Mining by graps / Trading wines
Hashmask [31] | Ethereum | Digital arts / Blind boxes
Cards | Sorare [58] | Ethereum | Football star cards
Skyweaver [56] | Ethereum | Card game
Cometh [16] | Ethereum | Galaxy background
Gods Unchained [29] | Ethereum | Trading card game / Hearthstone
Topps MLB [60] | WAX [66] | Baseball cards
R Planet [53] | WAX | Red planet background
Raising Pets | CryptoKitties [19] | Ethereum | The first game with hype in Ethereum
CryptoCats [18] | Ethereum | Raising pets and sell with high prices
Axie Infinity [9] | Ethereum
Virtual World | Sandbox [3] | Ethereum |  Create, explore and trade in the virtual world owned by users, also make money by selling the properties.
Decentraland [1] | Ethereum
Sommnium Space [57] | Ethereum
Cryptovoxels [22] | Ethereum
Alien Worlds [6] | WAX
MyCryptoHeros [43] | Ethereum | Historical story / Japan made
Real Properties | Crypto stamp [17] | Ethereum | NFTs on stamps
Diamonds [47] | Ethereum | NFTs on diamonds
Icecap [32] | Ethereum
Wiv [68] | Ethereum | NFTs on wine
Art Market | SuperRare [59] |  - |  A marketplace to create, collect and trade unique, single-edition artworks. Every artwork is authentically created by artists, and tokenized as a crypto- collectible digital item.
Rarible [55] |  -
Cargo [13] |  -
Async Art [8] |  -
Nifty Gateway [48] |  -
KnownOrigin [33] |  -
Statistic Web | NonFungible [49] |  - |  Trace and monitor NFT-related projects or generic types of cryptocurrencies by providing timely statistical data to show trends.
DappRadar [24] |  -
NFT bank[45] |  -
DefiPulse [26] |  -
Coingecko [15] |  -
Exchanges | Cryptoslam [21] |  |  A marketplace to trade unique, single-edition NFT-based properties.
Opensea [50] |  -
Zora [69]) |  -

  * Sources from NonFungible, Cryptoslam, DefiPulse, Coingecko, and DappRadar.

## Appendix D. Analysis of Loot

Loot [34][35] is a type of NFT product that uses ERC protocol on Ethererum,
with totally $8,000$ entries ($7778$ for trading, $222$ for team incentives).
The project presents all their products in the form of TEXT, removing
functionalities of images, stats, or any representative formats. Users can
only see eight lines of sentences, describing the attributes of game gears
(e.g., weapons, armor, necklaces, Rings). In order to create scarcity, Loot
adds randomized prefixes (with $42\%$ probability) or suffixes ($8.7\%$) for
each line of attributes (see the following code, line 2), according to the
assigned tokenID from users. Users can only use Etherscan’s smart contract to
cast loot, rather than its official website.

Selected code in Loot project

⬇

1function pluck(tokenId, keyPrefix, sourceArray) internal view returns (string
memory) {

2 uint256 rand = random(string(abi.encodePacked(keyPrefix,
toString(tokenId))));

3 string memory output = sourceArray[rand % sourceArray.length];

4 uint256 greatness = rand % 21;

5 if (greatness > 14) {

6 output = string(abi.encodePacked(output, " ", suffixes[rand %
suffixes.length]));

7 }

8 if (greatness >= 19) {

9 string[2] memory name;

10 name[0] = namePrefixes[rand % namePrefixes.length];

11 name[1] = nameSuffixes[rand % nameSuffixes.length];

12 if (greatness == 19) {

13 output = string(abi.encodePacked(’"’, name[0], ’ ’, name[1], ’" ’,
output));

14 } else {

15 output = string(abi.encodePacked(’"’, name[0], ’ ’, name[1], ’" ’, output,
" +1"));

16 }

17 }

18 return output;

19}

As faced by other NFTs, Loot confronts many drawbacks either. Firstly, the
value of NFT is uncertain. Loot is still in the early stage of development in
the current stage. Despite the price continuing to rise, there is no lack of
hype. The real consensus from communities should be maintained for a long
period. Secondly, difficulty in minting and high gas cost retard its wide
participant. Loot does not have a front-end Dapps, which can only be minted in
the Web3 browser. Complex steps become a big challenge for newcomers.
Meanwhile, the high gas fee of operations in Ethereum keeps many people away
from this game. The trading or minting fees can reach up to tens of hundreds
of dollars for each operation. Thirdly, the extensibility needs further
improvements. The current version (experimental stage) merely covers eight
explicit attributes in the RPG game. Also, there is no space for on-top
applications. The only future might be to develop more attributes and slots
for sale.


[a]Alexander Rothkopf

# Towards symmetric discretization schemes via weak boundary conditions

###### Abstract

The Szymanzik improvement program for gauge theories is most commonly
implemented using forward finite difference corrections to the Wilson action.
Central symmetric schemes naively applied, suffer from a doubling of degrees
of freedom, identical to the well known fermion doubling phenomenon. And while
adding a complex Wilson term remedies the problem for fermions, it does not
easily transfer to real-valued gauge fields. In this talk I report on recent
progress in formulating symmetric discretization schemes for classical actions
of simple one-dimensional problems. They avoid doubling by exploiting the weak
imposition of initial/boundary conditions. Inspired by recent work in the
field of numerical analysis of partial differential equations, I construct a
regularized summation-by-parts finite difference operator using boundary data
based on affine coordinates. Application to a classical initial value problems
with second order derivatives are presented.

##  1 Motivation

Many physical systems of experimental interest are of finite extent, be it the
droplet of quark-gluon plasma created in the interior of a relativistic heavy-
ion collision [1] or a cavity which strongly couples the light field to an
electron [2, 3]. In each case weak-coupling methods fail and a lattice field
theory evaluation of observables is called for. A finite volume entails loss
of translational invariance, similar to when localized sources are explicitly
placed in a system, a scenario relevant for the study of e.g. quarkonium bound
states in extreme conditions [4]. The question I would thus like to address is
how to develop improved discretization schemes for systems without translation
invariance.

The historic starting point for the discretization of lattice gauge theory is
Wilson’s plaquette action [5],

$\displaystyle P^{1\times
1}_{\mu\nu,x}=U_{\mu,x}U_{\nu,x+a_{\mu}{\hat{\mu}}}U^{\dagger}_{\mu,x+a_{\nu}{\hat{\nu}}}U^{\dagger}_{\nu,x}=e^{ia_{\mu}a_{\nu}\tilde{F}_{\mu\nu,x}}+{\cal
O}(a^{2}),$ (1) $\displaystyle\tilde{F}_{\mu\nu}=\Delta^{\rm
F}_{\mu}A_{\nu,x}-\Delta^{\rm F}_{\nu}A_{\mu,x}+i[A_{\mu,x},A_{\nu,x}],$ (2)

which corresponds to a forward finite difference approximation $\Delta^{\rm
F}_{\mu}\phi(x)=(\phi(x+a_{\mu}\hat{\mu})-\phi(x))/a_{\mu}$ of the field
strength tensor. The associated Gauss law turns out to be described by a
backward finite difference (BFD) operator. The limitations of such a BFD
discretization are already visible on the level of classical electrodynamics.
Inspired by [6] I discussed in [7] two relevant Abelian model systems: a
capacitor with a finite charge density on its plates, as well the as a charge-
anticharge pair. Prescribing the true values of the electric field on the
boundary plates, the BFD discretization $\Delta^{B}\cdot{\bf E}=0$ manages to
sustain field strength in the interior of the capacitor only close to the
backward boundary. As shown on the left of fig. 1 the field values on the
forward facing plate are invisible to the BFD operator. When solving Gauss’
law in the presence of a charge anticharge pair (right panel of fig. 1) and
computing the field lines in a gauge invariant fashion by diagonalizing the
stress tensor (blue arrows), one sees that they show a significant imbalance
towards the backward direction.

Figure 1: (left) field lines from solving Gauss’ law in the interior of a
capacitor for prescribed values of the true electric field on the boundary
(plates). Due to the backward finite difference prescription, field strength
is only sustained close to the backward boundary. (right) The field lines
arising from solving the BFD Gauss law in the presence of a charge anticharge
pair described by point sources (true solution in light red).

The gauge actions deployed in the lattice QCD community today are improved, in
the sense of the Szymanzik program [8]. To accelerate the approach to the
continuum limit, i.e. to reduce discretization artifacts in the simulated
correlators, the plaquettes of the Wilson action are amended by loops with
larger than unit area. Most common correction terms are the forward
rectangles. And while these terms indeed reduce the lattice spacing dependence
of the simulation they do not realize a symmetric discretization of the field
strength around charges. Recovering the Gauss law in the continuum limit
therefore becomes non-trivial.

To realize a genuine symmetric discretization scheme, I proposed in [7] to use
instead the $2\times 2$ plaquette centered around the nodes of the grid

$\displaystyle P^{2\times 2}_{\mu\nu,x}=$
$\displaystyle\bar{U}_{\mu,x-a\hat{\mu}-a\hat{\nu}}\bar{U}_{\mu,x-a\hat{\nu}}\bar{U}_{\nu,x+a\hat{\mu}-a\hat{\nu}}\bar{U}_{\nu,x+a\hat{\mu}}\bar{U}^{\dagger}_{\mu,x+a\hat{\nu}}\bar{U}^{\dagger}_{\mu,x-a\hat{\mu}+a\hat{\nu}}\bar{U}^{\dagger}_{\nu,x-a\hat{\mu}}\bar{U}^{\dagger}_{\nu,x-a\hat{\mu}-a\hat{\nu}}$
$\displaystyle=$ $\displaystyle{\rm
exp}\big{[}4iga_{\mu}a_{\nu}\bar{F}_{\mu\nu,x}\big{]}+{\cal
O}(a^{3}),\quad\bar{F}_{\mu\nu,x}={\bf\Delta}^{\rm
C}_{\mu}A_{\nu,x}-{\bf\Delta}^{\rm C}_{\nu}A_{\mu,x}+i[A_{\mu,x},A_{\nu,x}]$

which realizes a central finite difference discretization $\Delta^{\rm
C}_{\mu}\phi(x)=(\phi(x+a_{\mu}\hat{\mu})-\phi(x-a_{\mu}\hat{\mu}))/2a_{\mu}$
of the field strength. This plaquette differs from the usual clover leaf
prescription in that it corresponds to the product of four $1\times 1$
plaquettes and thus remains within the gauge group (see also its relation to
stabilized Wilson fermions [9]).

First simulations [10] based on the action $S^{2\times
2}=\sum_{x}a_{t}a_{s}^{3}\Big{[}\frac{2}{16a_{t}^{2}a_{s}^{2}}\sum_{i}{\rm
ReTr}\big{[}1-P^{2\times 2}_{0i,x}\big{]}-\frac{1}{16a_{s}^{4}}\sum_{ij}{\rm
ReTr}\big{[}1-P^{2\times 2}_{ij,x}\big{]}\Big{]}$ however revealed (see fig.
2) that such a symmetric discretization scheme for gauge fields suffers from
bosonic doublers, similar to the ones encountered when using symmetric finite
differences in the discretization of fermion fields. I.e. the values of the
trace anomaly in the right panel of fig. 2 are larger than the correct result
by a factor of $4\times 8\times 15$ (c.f. [11]), indicating that for each
physical mode, fifteen unphysical degrees of freedom propagate in the
interior.

Figure 2: (left) Renormalized sum of the spatial and temporal $P^{2\times 2}$
plaquettes that enter the computation of the trace anomaly in simulations with
$S^{2\times 2}$. (right) Values of the trace anomaly from naively implemented
$S^{2\times 2}$, which is larger than the correct value by a factor of
$4\times 8\times 15$.

##  2 Regularization with boundary data

Let us illustrate the doubling problem in coordinate space in one dimension,
using the following finite difference operator $D^{C}$ with symmetric stencil
in the interior

$\displaystyle D^{\rm C}=\frac{1}{\Delta
x}\left[\begin{array}[]{cccc}-1&1&0&0\\\ -\frac{1}{2}&0&\frac{1}{2}&0\\\
0&-\frac{1}{2}&0&\frac{1}{2}\\\
0&0&-1&1\end{array}\right],\qquad\tilde{D}=D^{C}+\kappa\frac{\Delta x}{2\Delta
x^{2}}\left[\begin{array}[]{cccc}\ddots&1&0&0\\\ 1&-2&1&0\\\ 0&1&-2&1\\\
0&0&1&\ddots\end{array}\right].$ (11)

The operator $D^{C}$ is a so-called summation by parts operator, as it mimics
accurately integration by parts in the discrete setting. (see e.g. [12] and
for the related discussion of the momentum operator on finite domains see [13,
14]). If we compute its spectrum, we find that it features two zero
eigenvalues. They are associated with degenerate eigenfunctions that turn out
to be the constant function (blue circles in the left panel of fig. 3). The
doublers are hiding in the eigenfunctions of the transpose $(D^{C})^{t}$ and
are indeed the maximally oscillating function on the grid also with eigenvalue
zero (green triangles).

Figure 3: (left) The degenerate eigenfunctions associated with the zero
eigenvalues of $D^{C}$ (blue circles) and with its transpose $(D^{C})^{t}$
(green triangles). (right) Eigenvalue spectrum of the operator $D^{C}$ with
exact zero modes (blue circles) and of the regularized operator $\tilde{D}$
(green triangles).

In the numerical analysis literature several regularization approaches are
discussed. One of them, called the upwind modification ($\kappa=1$ in eq. 11
), adds a symmetric second derivative to $D^{C}$ with one extra power of the
grid spacing $\Delta x$. This term vanishes in the continuum limit and does
not affect the defining property of the finite difference, i.e. $\tilde{D}{\bf
x}^{r}=r{\bf x}^{r-1}$ for $r$ smaller than the order of $D^{C}$. At the same
time this modification destroys the symmetry of the interior stencil and
reverts the derivative to the naive forward form. It was Wilson’s seminal
contribution [5] to realize that for complex valued fermionic fields the
higher order derivative can be added with an imaginary prefactor ($\kappa=i$
in eq. 11) which retains the symmetry of both terms while lifting the zero
modes. This is the celebrated Wilson term. In case of bosonic gauge fields,
which must remain real-valued, a similar modification however is not possible.

Taking inspiration from the computational fluid dynamics community, I propose
to use another lever to regularize the bosonic doubler problem: boundary
values. In a finite system boundary data are physical information and
otherwise can be chosen at convenience. In the past, boundary conditions were
mostly implemented in the strong sense, i.e one replaces the degrees of
freedom on the boundary apriori with the prescribed values. Alternatively one
may consider the weak treatment of boundary conditions, which acknowledges
that the boundary conditions only need to be realized as accurately as the
rest of the discretization. In turn one can introduce boundary information
through penalty terms, which offer new opportunities for regularization. This
approach, known as simultaneous approximation terms (SAT) [15] is well
established in the treatment of classical ODEs and PDEs [16].

Consider the continuum problem of solving the ODE $u^{\prime}(x)=g(x)$ with
boundary condition $u(0)=u_{0}$. In discretized form it can be formulated as

$\displaystyle D{\bf u}={\bf g}+\frac{1}{\Delta x}E_{0}\big{(}{\bf u}-{\bf
u}_{0}\big{)}\quad\rightarrow\quad\tilde{D}=D-\frac{1}{\Delta
x}E_{0}\quad\rightarrow\quad\quad\tilde{D}{\bf u}={\bf g}-\frac{1}{\Delta
x}E_{0}{\bf u}_{0}.$ (12)

where we use the notation ${\bf u}_{k}=u(k\Delta x)$ and introduce a penalty
term that refers to the boundary values via the projection matrix $E_{0}={\rm
diag}[1,0,\ldots]$ and ${\bf u}_{0}=\\{u_{0},0,\ldots\\}$. As the lattice
spacing is reduced the penalty increases and the boundary conditions will be
more strictly fulfilled. The form of the penalty term invites us to absorb its
homogeneous part into a redefined difference operator $\tilde{D}$. In the
right panel of fig. 3 we show the effect of this redefinition on the
eigenvalues of the finite difference operator. The blue points represent the
eigenvalues of $D$, which feature two exact zeros connected to the doubler
mode. The green points on the other hand denote the eigenvalues of $\tilde{D}$
where all zero modes have been lifted. Thus $\tilde{D}$ is an invertible
operator leading to a unique solution of the system of equations for ${\bf u}$
in the right-most term in eq. 12.

On the lattice we need to incorporate this regularization in the action of the
system. The difficulty here lies in the fact that now we do not have an equal
sign to move the penalty term around, as we did in eq. 12. Together with J.
Nordstöm, I recently proposed a solution by incorporating the penalty term as
a whole in the definition of the finite difference operator using affine
coordinates. Take as example the continuum action $S=\int
dxu^{\prime}(x)u^{\prime}(x)$ with some boundary condition $u(0)=u_{0}$.
Implementing the quadrature of the integral using a matrix (e.g. $H=\Delta
x{\rm diag}[\frac{1}{2},1,\ldots,1,\frac{1}{2}]$ for the trapezoid rule) we
can write $S\approx(D{\bf u})^{t}HD{\bf u}$. Our proposal is to define a new
regularized finite difference operator in which the boundary penalty term is
included

$\displaystyle\bar{D}{\bf u}=D{\bf u}+H^{-1}E_{0}\big{(}{\bf u}-{\bf
u}_{0}\big{)},\qquad\bar{D}=\frac{1}{\Delta
x}\left[\begin{array}[]{ccccc}-1+2&1&0&0&-2u_{0}\\\
-\frac{1}{2}&0&\frac{1}{2}&0&0\\\ 0&-\frac{1}{2}&0&\frac{1}{2}&0\\\
0&0&-1&1&0\\\ 0&0&0&0&1\end{array}\right].$ (18)

The last term involving ${\bf u}_{0}$ is nothing but a shift, which can be
conveniently included in matrix form when amending the matrix of $\bar{D}$ by
one row and one column, placing the value one in the lower right corner and
filling the extra column on the right with the values of the shift (see term
on the right in eq. 18). For consistency all vectors corresponding to
discretized functions are also amended with one more entry of the value one
$\tilde{\bf u}=\\{{\bf u},1\\}$. The absorption of the boundary term into
$\bar{D}$ has a similar effect on its eigenvalues as we observed for the
regularized $\tilde{D}$ previously. Both zero modes are lifted. For $\bar{D}$
expressed in affine coordinates we find that there exists a single purely real
eigenvalue of value one, while all others come in complex conjugate pairs.
This eigenvalue is now associated with the zero function ${\bf u}^{(0)}_{\rm
aff}=\\{0,\ldots,0,1\\}$ in the spectrum of $\bar{D}^{t}$, from which the
maximally oscillating doubler mode has been deleted.

The inclusion of boundary data as shown here, constitutes a novel
regularization procedure for symmetric discretization schemes. It is
applicable also to purely real field degrees of freedom, in contrast to the
Wilson term, which requires complex valued functions.

##  3 Application to simple initial value problems

As a first application of the regularization procedure it has been used in
[17] to develop a novel discretization prescription for classical initial
value problems (IVP), based on a variational principle. The long-term goal of
this line of study is to realize genuine real-time simulations of quantum
fields on the lattice (see e.g. our work on complex Langevin [18] ). However
as an intermediate time goal I see the realization of gauge invariant
simulations of the real-time dynamics of classical lattice gauge theory. Today
these simulations are based on Hamilton’s equations of motion, which require a
choice of gauge. The first modest step in this direction I am going to discuss
here is the development of a variational solver for initial value problems in
classical point mechanics.

Figure 4: Sketches of (left) the conventional boundary value formulation of
the variational principle in classical mechanics and (right) the variational
principle adapted to initial value problems, which requires a doubling of
degrees of freedom.

The challenge one faces with the construction of variational solvers for IVPs
is the fact that conventionally the variational problem is formulated as a
boundary value problem (see left panel of fig. 4). This is unsatisfactory, as
the position of the system at final time $t_{2}$ needs to be provided apriori,
an information that is not available in a genuine IVP. As was shown in [19] a
variational principle can be established when introducing a doubling of the
degrees of freedom, leading in essence to a double shooting method. In essence
[19] discusses the classical limit of the Schwinger-Keldysh real-time contour
prescription, where in addition to degrees of freedom on the forward branch
$x_{1}$ also $x_{2}$ on a backward time branch are considers. The latter are
assigned a negative weight in the system action.

$\displaystyle{\cal S}_{\rm
IVP}[x_{1}(t),\dot{x}_{1}(t),x_{2}(t),\dot{x}_{2}(t)]$
$\displaystyle=\int_{t_{1}}^{t_{2}}dt\Big{(}{\cal
L}[x_{1}(t),\dot{x}_{1}(t)]-{\cal L}[x_{2}(t),\dot{x}_{2}(t)]\Big{)},$ (19)
$\displaystyle=\int_{t_{1}}^{t_{2}}dt\Big{(}{\rm
L}[x_{1}(t),\dot{x}_{1}(t),x_{2}(t),\dot{x}_{2}(t)]\Big{)}.$ (20)

Let us introduce the coordinates $x_{+}=(x_{1}+x_{2})/2$ and
$x_{-}=x_{1}-x_{2}$, so that the variation of the system action can be
expressed in the form

$\displaystyle\delta{\cal S}=\int dt\Big{(}\Big{\\{}\frac{\partial{\rm
L}}{\partial x_{1}}-\frac{d}{dt}\frac{\partial{\rm
L}}{\partial\dot{x}_{1}}\Big{\\}}\delta x_{1}-\Big{\\{}\frac{\partial{\rm
L}}{\partial x_{2}}-\frac{d}{dt}\frac{\partial{\rm
L}}{\partial\dot{x}_{2}}\Big{\\}}\delta
x_{2}\Big{)}+\left.\Big{[}\frac{\partial{\rm L}}{\partial\dot{x}_{1}}\delta
x_{1}\Big{]}\right|_{t_{1}}^{t_{2}}-\left.\Big{[}\frac{\partial{\rm
L}}{\partial\dot{x}_{2}}\delta x_{2}\Big{]}\right|_{t_{1}}^{t_{2}}.$ (21)

Now if one enforces that the value of the paths and their derivatives agree at
the final time step $t_{2}$ one can show that the critical point of the action
is equivalent to the solution of the Euler-Lagrange equations. I.e. even
though the value of the classical path at the final time step $t_{2}$ is not
fixed to a certain value, the contributions from $x_{1}(t_{2})$ and
$x_{2}(t_{2})$ are designed such that they cancel correctly the boundary terms
arising there. The stationarity condition arising from this variational
principle is most concisely formulated as $\left.\frac{\delta S_{\rm
IVP}[x_{\pm}]}{\delta x_{-}}\right|_{x_{-}=0,x_{+}=x_{\rm class}}=0$. The
above derivation did not make any reference to quantum field theory, but
reassuringly reproduces the result of [20] where the classical limit of the
Schwinger-Keldysh formalism was investigated.

We set out to discretize and solve the variational principle for a simple IVP,
the point particle in a constant gravitational field with continuum action
${\cal S}=\int\,dt\Big{(}\frac{1}{2}m\dot{x}^{2}(t)-mgx(t)\Big{)}$ and
arbitrarily chosen initial conditions $x(0)=1$ and $\dot{x}(0)=0.3$. According
to eq. 20 we must double the degrees of freedom, hence we introduce ${\bf
x}_{1}$ and ${\bf x}_{2}$. The naive discretization of the action with
quadrature matrix $H$ and the naive SBP finite difference operator $D^{C}$
introduced in eq. 11 reads

$\displaystyle{\cal S}_{\rm IVP}=\Big{\\{}\frac{1}{2}(D^{C}{\bf x}_{1})^{\rm
T}{H}(D^{C}{\bf x}_{1})-g{1}^{\rm T}{H}{\bf
x}_{1}\Big{\\}}-\Big{\\{}\frac{1}{2}(D^{C}{\bf x}_{2})^{\rm T}{H}(D^{C}{\bf
x}_{2})-g{1}^{\rm T}{H}{\bf x}_{2}\Big{\\}}$ (22)
$\displaystyle+\lambda_{1}(x_{1}(0)-x_{i})+\lambda_{2}((D^{C}{\bf
x}_{1})(0)-\dot{x}_{i})+\lambda_{3}(x_{1}(N_{t})-x_{2}(N_{t}))+\lambda_{4}((D^{C}{\bf
x}_{1})(N_{t})-(D^{C}{\bf x}_{2})(N_{t})).$

In order to locate the critical point of the action under the constraints that
initial conditions shall be fulfilled and that the paths ${\bf x}_{1}$ and
${\bf x}_{2}$ shall agree at the last time point we have added four Langrange
multipliers $\lambda_{i}$. Using $N_{t}=32$ steps and a time extent of
$t_{2}=1$, we carry out a numerical minimization of the expression in eq. 22
which leads to the result shown in the left panel of fig. 5. We find that the
obtained solution (crosses for ${\bf x}_{1}$, circles for ${\bf x}_{2}$)
fulfills the requirement ${\bf x}_{1}={\bf x}_{2}$ of the stationarity
condition. One can however clearly see that only a subset of the solution lies
on the correct solution (gray solid line) and that a significant portion of
the points on the paths form an artificial oscillatory pattern. From our
investigation of the spectrum of $D^{C}$ this is not surprising, as the
maximally oscillating zero mode, the doubler, has simply contaminated the
result.

If instead we use the finite difference operator $\bar{D}$ in affine
coordinates, which is regularized using the initial value data, together with
the corresponding paths $\bar{\bf x}_{1,2}$ and quadrature matrix $\bar{H}$ in
affine coordinates, we end up with the following expression

$\displaystyle{\cal S}^{\rm reg}_{\rm
IVP}=\Big{\\{}\frac{1}{2}(\bar{D}\bar{\bf x}_{1})^{\rm
T}{\bar{H}}(\bar{D}\bar{\bf x}_{1})-g{1}^{\rm T}{H}{\bf
x}_{1}\Big{\\}}-\Big{\\{}\frac{1}{2}(\bar{D}\bar{\bf x}_{2})^{\rm
T}{\bar{H}}(\bar{D}\bar{\bf x}_{2})-g{1}^{\rm T}{H}{\bf x}_{2}\Big{\\}}$ (23)
$\displaystyle+\lambda_{1}(x_{1}(0)-x_{i})+\lambda_{2}((D^{C}{\bf
x}_{1})(0)-\dot{x}_{i})+\lambda_{3}(x_{1}(N_{t})-x_{2}(N_{t}))+\lambda_{4}((D^{C}{\bf
x}_{1})(N_{t})-(D^{C}{\bf x}_{2})(N_{t})).$

whose critical trajectory is shown on the right of fig. 5. The regularization
has successfully avoided the occurrence of doublers and manages to bring the
solution close to the correct classical trajectory. Of course the accuracy of
the solution is limited by the accuracy of the discretization used for the
finite difference operator $\bar{D}$. Systematic prescriptions for the
construction of higher order SBP operators exist and the regularization in
affine coordinates is independent of the form of $D$, i.e. it can be applied
straight forwardly to higher order operators too. We have checked that the
classical trajectory found via this variational approach correctly approaches
the continuum limit under grid refinement (for a detailed analysis see [17]).

Figure 5: (left) Solution of the IVP of eq. 22 with a naive SBP operator which
suffers from the occurrence of doublers. (right) Solution of the IVP of eq. 23
with the regularized SBP operator in affine coordinates which avoids doublers.
Note that the solutions for ${\bf x}_{1}$ (crosses) and ${\bf x}_{2}$
(circles) agree as required by the stationarity condition.

##  4 Conclusion

The physics of strongly correlated fields in finite systems and in the
presence of explicit sources requires discretization schemes in the absence of
translation invariance. A proposal to deploy the centrally symmetric $2\times
2$ plaquette revealed the occurrence of bosonic doublers in such symmetric
discretization schemes. The challenge lies in the fact that the Wilson term
regularization, successful for complex valued fermion fields is not applicable
for real-valued gauge fields. Instead I propose to use the weak imposition of
boundary data as an alternative regularization mechanism, which can be
straight forwardly implemented when expressing finite difference operators in
affine coordinates. As a first step, the efficacy of this type of
regularization has been demonstrated in a variational solver for classical
initial value problems. While unregularized finite difference operators lead
to solutions for the classical trajectory that are contaminated by doubler
modes, the regularized operator successfully avoids the occurrence of the
bosonic doublers. Extension of the variational approach to higher dimensions
is work in progress.

##  5 Acknowledgments

A.R. is supported by the Research Council of Norway under the FRIPRO Young
Research Talent grant 286883.

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